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Theorem islindf4 19996
 Description: A family is independent iff it has no nontrivial representations of zero. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Hypotheses
Ref Expression
islindf4.b 𝐵 = (Base‘𝑊)
islindf4.r 𝑅 = (Scalar‘𝑊)
islindf4.t · = ( ·𝑠𝑊)
islindf4.z 0 = (0g𝑊)
islindf4.y 𝑌 = (0g𝑅)
islindf4.l 𝐿 = (Base‘(𝑅 freeLMod 𝐼))
Assertion
Ref Expression
islindf4 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0𝑥 = (𝐼 × {𝑌}))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐹   𝑥,𝐼   𝑥,𝐿   𝑥,𝑅   𝑥, ·   𝑥,𝑊   𝑥,𝑋   𝑥,𝑌   𝑥, 0

Proof of Theorem islindf4
Dummy variables 𝑗 𝑘 𝑙 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 raldifsni 4265 . . . . 5 (∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ (Base‘𝑅)((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌))
2 simpll1 1093 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ LMod)
3 simprll 798 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑙 ∈ (Base‘𝑅))
4 ffvelrn 6265 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝐼𝐵𝑗𝐼) → (𝐹𝑗) ∈ 𝐵)
543ad2antl3 1218 . . . . . . . . . . . . . . . . 17 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (𝐹𝑗) ∈ 𝐵)
65adantr 480 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝐹𝑗) ∈ 𝐵)
7 islindf4.b . . . . . . . . . . . . . . . . 17 𝐵 = (Base‘𝑊)
8 islindf4.r . . . . . . . . . . . . . . . . 17 𝑅 = (Scalar‘𝑊)
9 islindf4.t . . . . . . . . . . . . . . . . 17 · = ( ·𝑠𝑊)
10 eqid 2610 . . . . . . . . . . . . . . . . 17 (invg𝑊) = (invg𝑊)
11 eqid 2610 . . . . . . . . . . . . . . . . 17 (invg𝑅) = (invg𝑅)
12 eqid 2610 . . . . . . . . . . . . . . . . 17 (Base‘𝑅) = (Base‘𝑅)
137, 8, 9, 10, 11, 12lmodvsinv 18857 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘𝑅) ∧ (𝐹𝑗) ∈ 𝐵) → (((invg𝑅)‘𝑙) · (𝐹𝑗)) = ((invg𝑊)‘(𝑙 · (𝐹𝑗))))
142, 3, 6, 13syl3anc 1318 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((invg𝑅)‘𝑙) · (𝐹𝑗)) = ((invg𝑊)‘(𝑙 · (𝐹𝑗))))
1514eqeq1d 2612 . . . . . . . . . . . . . 14 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ ((invg𝑊)‘(𝑙 · (𝐹𝑗))) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))))
16 lmodgrp 18693 . . . . . . . . . . . . . . . 16 (𝑊 ∈ LMod → 𝑊 ∈ Grp)
172, 16syl 17 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ Grp)
187, 8, 9, 12lmodvscl 18703 . . . . . . . . . . . . . . . 16 ((𝑊 ∈ LMod ∧ 𝑙 ∈ (Base‘𝑅) ∧ (𝐹𝑗) ∈ 𝐵) → (𝑙 · (𝐹𝑗)) ∈ 𝐵)
192, 3, 6, 18syl3anc 1318 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑙 · (𝐹𝑗)) ∈ 𝐵)
20 islindf4.z . . . . . . . . . . . . . . . 16 0 = (0g𝑊)
21 lmodcmn 18734 . . . . . . . . . . . . . . . . 17 (𝑊 ∈ LMod → 𝑊 ∈ CMnd)
222, 21syl 17 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ CMnd)
23 simpll2 1094 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐼𝑋)
24 difexg 4735 . . . . . . . . . . . . . . . . 17 (𝐼𝑋 → (𝐼 ∖ {𝑗}) ∈ V)
2523, 24syl 17 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝐼 ∖ {𝑗}) ∈ V)
26 simprlr 799 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))
27 elmapi 7765 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → 𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅))
2826, 27syl 17 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅))
29 simpll3 1095 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹:𝐼𝐵)
30 difss 3699 . . . . . . . . . . . . . . . . . 18 (𝐼 ∖ {𝑗}) ⊆ 𝐼
31 fssres 5983 . . . . . . . . . . . . . . . . . 18 ((𝐹:𝐼𝐵 ∧ (𝐼 ∖ {𝑗}) ⊆ 𝐼) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
3229, 30, 31sylancl 693 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
338, 12, 9, 7, 2, 28, 32, 25lcomf 18725 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))):(𝐼 ∖ {𝑗})⟶𝐵)
34 islindf4.y . . . . . . . . . . . . . . . . 17 𝑌 = (0g𝑅)
35 simprr 792 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦 finSupp 𝑌)
368, 12, 9, 7, 2, 28, 32, 25, 20, 34, 35lcomfsupp 18726 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))) finSupp 0 )
377, 20, 22, 25, 33, 36gsumcl 18139 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ∈ 𝐵)
38 eqid 2610 . . . . . . . . . . . . . . . 16 (+g𝑊) = (+g𝑊)
397, 38, 20, 10grpinvid2 17294 . . . . . . . . . . . . . . 15 ((𝑊 ∈ Grp ∧ (𝑙 · (𝐹𝑗)) ∈ 𝐵 ∧ (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ∈ 𝐵) → (((invg𝑊)‘(𝑙 · (𝐹𝑗))) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ ((𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g𝑊)(𝑙 · (𝐹𝑗))) = 0 ))
4017, 19, 37, 39syl3anc 1318 . . . . . . . . . . . . . 14 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((invg𝑊)‘(𝑙 · (𝐹𝑗))) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ ((𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g𝑊)(𝑙 · (𝐹𝑗))) = 0 ))
41 simplr 788 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑗𝐼)
42 fsnunf2 6357 . . . . . . . . . . . . . . . . . . 19 ((𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) ∧ 𝑗𝐼𝑙 ∈ (Base‘𝑅)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}):𝐼⟶(Base‘𝑅))
4328, 41, 3, 42syl3anc 1318 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}):𝐼⟶(Base‘𝑅))
448, 12, 9, 7, 2, 43, 29, 23lcomf 18725 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹):𝐼𝐵)
45 simpr 476 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝑗𝐼)
46 simpl 472 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) → 𝑙 ∈ (Base‘𝑅))
4745, 46anim12i 588 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (𝑗𝐼𝑙 ∈ (Base‘𝑅)))
48 elmapfun 7767 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → Fun 𝑦)
49 fdm 5964 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) → dom 𝑦 = (𝐼 ∖ {𝑗}))
50 neldifsnd 4263 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (dom 𝑦 = (𝐼 ∖ {𝑗}) → ¬ 𝑗 ∈ (𝐼 ∖ {𝑗}))
51 df-nel 2783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑗 ∉ dom 𝑦 ↔ ¬ 𝑗 ∈ dom 𝑦)
52 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (dom 𝑦 = (𝐼 ∖ {𝑗}) → (𝑗 ∈ dom 𝑦𝑗 ∈ (𝐼 ∖ {𝑗})))
5352notbid 307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (dom 𝑦 = (𝐼 ∖ {𝑗}) → (¬ 𝑗 ∈ dom 𝑦 ↔ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})))
5451, 53syl5bb 271 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (dom 𝑦 = (𝐼 ∖ {𝑗}) → (𝑗 ∉ dom 𝑦 ↔ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})))
5550, 54mpbird 246 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (dom 𝑦 = (𝐼 ∖ {𝑗}) → 𝑗 ∉ dom 𝑦)
5649, 55syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) → 𝑗 ∉ dom 𝑦)
5727, 56syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → 𝑗 ∉ dom 𝑦)
5848, 57jca 553 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})) → (Fun 𝑦𝑗 ∉ dom 𝑦))
5958adantl 481 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) → (Fun 𝑦𝑗 ∉ dom 𝑦))
6059adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (Fun 𝑦𝑗 ∉ dom 𝑦))
6147, 60jca 553 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → ((𝑗𝐼𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦𝑗 ∉ dom 𝑦)))
62 funsnfsupp 8182 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑗𝐼𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦𝑗 ∉ dom 𝑦)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌𝑦 finSupp 𝑌))
6362bicomd 212 . . . . . . . . . . . . . . . . . . . . 21 (((𝑗𝐼𝑙 ∈ (Base‘𝑅)) ∧ (Fun 𝑦𝑗 ∉ dom 𝑦)) → (𝑦 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
6461, 63syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (𝑦 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
6564biimpd 218 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (𝑦 finSupp 𝑌 → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
6665impr 647 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌)
678, 12, 9, 7, 2, 43, 29, 23, 20, 34, 66lcomfsupp 18726 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) finSupp 0 )
68 incom 3767 . . . . . . . . . . . . . . . . . . 19 ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ({𝑗} ∩ (𝐼 ∖ {𝑗}))
69 disjdif 3992 . . . . . . . . . . . . . . . . . . 19 ({𝑗} ∩ (𝐼 ∖ {𝑗})) = ∅
7068, 69eqtri 2632 . . . . . . . . . . . . . . . . . 18 ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ∅
7170a1i 11 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝐼 ∖ {𝑗}) ∩ {𝑗}) = ∅)
72 difsnid 4282 . . . . . . . . . . . . . . . . . . 19 (𝑗𝐼 → ((𝐼 ∖ {𝑗}) ∪ {𝑗}) = 𝐼)
7372eqcomd 2616 . . . . . . . . . . . . . . . . . 18 (𝑗𝐼𝐼 = ((𝐼 ∖ {𝑗}) ∪ {𝑗}))
7441, 73syl 17 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐼 = ((𝐼 ∖ {𝑗}) ∪ {𝑗}))
757, 20, 38, 22, 23, 44, 67, 71, 74gsumsplit 18151 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = ((𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})))(+g𝑊)(𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}))))
76 vex 3176 . . . . . . . . . . . . . . . . . . . . 21 𝑦 ∈ V
77 snex 4835 . . . . . . . . . . . . . . . . . . . . 21 {⟨𝑗, 𝑙⟩} ∈ V
7876, 77unex 6854 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∈ V
79 simpl3 1059 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐹:𝐼𝐵)
80 simpl2 1058 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐼𝑋)
81 fex 6394 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:𝐼𝐵𝐼𝑋) → 𝐹 ∈ V)
8279, 80, 81syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐹 ∈ V)
8382adantr 480 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹 ∈ V)
84 offres 7054 . . . . . . . . . . . . . . . . . . . 20 (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∈ V ∧ 𝐹 ∈ V) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
8578, 83, 84sylancr 694 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
86 ffn 5958 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦:(𝐼 ∖ {𝑗})⟶(Base‘𝑅) → 𝑦 Fn (𝐼 ∖ {𝑗}))
8728, 86syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑦 Fn (𝐼 ∖ {𝑗}))
88 neldifsn 4262 . . . . . . . . . . . . . . . . . . . . 21 ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})
89 fsnunres 6359 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦 Fn (𝐼 ∖ {𝑗}) ∧ ¬ 𝑗 ∈ (𝐼 ∖ {𝑗})) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) = 𝑦)
9087, 88, 89sylancl 693 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) = 𝑦)
9190oveq1d 6564 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ↾ (𝐼 ∖ {𝑗})) ∘𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))) = (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
9285, 91eqtrd 2644 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})) = (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))
9392oveq2d 6565 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗}))) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))))
94 ffn 5958 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹):𝐼𝐵 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) Fn 𝐼)
9544, 94syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) Fn 𝐼)
96 fnressn 6330 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) Fn 𝐼𝑗𝐼) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}) = {⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗)⟩})
9795, 41, 96syl2anc 691 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}) = {⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗)⟩})
98 ffn 5958 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 ∪ {⟨𝑗, 𝑙⟩}):𝐼⟶(Base‘𝑅) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼)
9943, 98syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼)
100 ffn 5958 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝐹:𝐼𝐵𝐹 Fn 𝐼)
10129, 100syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝐹 Fn 𝐼)
102 fnfvof 6809 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑦 ∪ {⟨𝑗, 𝑙⟩}) Fn 𝐼𝐹 Fn 𝐼) ∧ (𝐼𝑋𝑗𝐼)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) · (𝐹𝑗)))
10399, 101, 23, 41, 102syl22anc 1319 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗) = (((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) · (𝐹𝑗)))
104 fndm 5904 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 Fn (𝐼 ∖ {𝑗}) → dom 𝑦 = (𝐼 ∖ {𝑗}))
105104eleq2d 2673 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 Fn (𝐼 ∖ {𝑗}) → (𝑗 ∈ dom 𝑦𝑗 ∈ (𝐼 ∖ {𝑗})))
10688, 105mtbiri 316 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 Fn (𝐼 ∖ {𝑗}) → ¬ 𝑗 ∈ dom 𝑦)
107 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑗 ∈ V
108 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑙 ∈ V
109 fsnunfv 6358 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑗 ∈ V ∧ 𝑙 ∈ V ∧ ¬ 𝑗 ∈ dom 𝑦) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
110107, 108, 109mp3an12 1406 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑗 ∈ dom 𝑦 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
11187, 106, 1103syl 18 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑙)
112111oveq1d 6564 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) · (𝐹𝑗)) = (𝑙 · (𝐹𝑗)))
113103, 112eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗) = (𝑙 · (𝐹𝑗)))
114113opeq2d 4347 . . . . . . . . . . . . . . . . . . . . 21 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗)⟩ = ⟨𝑗, (𝑙 · (𝐹𝑗))⟩)
115114sneqd 4137 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → {⟨𝑗, (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)‘𝑗)⟩} = {⟨𝑗, (𝑙 · (𝐹𝑗))⟩})
116 ovex 6577 . . . . . . . . . . . . . . . . . . . . . 22 (𝑙 · (𝐹𝑗)) ∈ V
117 fmptsn 6338 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑗 ∈ V ∧ (𝑙 · (𝐹𝑗)) ∈ V) → {⟨𝑗, (𝑙 · (𝐹𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗))))
118107, 116, 117mp2an 704 . . . . . . . . . . . . . . . . . . . . 21 {⟨𝑗, (𝑙 · (𝐹𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗)))
119118a1i 11 . . . . . . . . . . . . . . . . . . . 20 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → {⟨𝑗, (𝑙 · (𝐹𝑗))⟩} = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗))))
12097, 115, 1193eqtrd 2648 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}) = (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗))))
121120oveq2d 6565 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗})) = (𝑊 Σg (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗)))))
122 cmnmnd 18031 . . . . . . . . . . . . . . . . . . . 20 (𝑊 ∈ CMnd → 𝑊 ∈ Mnd)
12322, 122syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑊 ∈ Mnd)
124107a1i 11 . . . . . . . . . . . . . . . . . . 19 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑗 ∈ V)
125 eqidd 2611 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑗 → (𝑙 · (𝐹𝑗)) = (𝑙 · (𝐹𝑗)))
1267, 125gsumsn 18177 . . . . . . . . . . . . . . . . . . 19 ((𝑊 ∈ Mnd ∧ 𝑗 ∈ V ∧ (𝑙 · (𝐹𝑗)) ∈ 𝐵) → (𝑊 Σg (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗)))) = (𝑙 · (𝐹𝑗)))
127123, 124, 19, 126syl3anc 1318 . . . . . . . . . . . . . . . . . 18 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (𝑥 ∈ {𝑗} ↦ (𝑙 · (𝐹𝑗)))) = (𝑙 · (𝐹𝑗)))
128121, 127eqtrd 2644 . . . . . . . . . . . . . . . . 17 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗})) = (𝑙 · (𝐹𝑗)))
12993, 128oveq12d 6567 . . . . . . . . . . . . . . . 16 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ (𝐼 ∖ {𝑗})))(+g𝑊)(𝑊 Σg (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹) ↾ {𝑗}))) = ((𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g𝑊)(𝑙 · (𝐹𝑗))))
13075, 129eqtr2d 2645 . . . . . . . . . . . . . . 15 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g𝑊)(𝑙 · (𝐹𝑗))) = (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)))
131130eqeq1d 2612 . . . . . . . . . . . . . 14 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))(+g𝑊)(𝑙 · (𝐹𝑗))) = 0 ↔ (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 ))
13215, 40, 1313bitrd 293 . . . . . . . . . . . . 13 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) ↔ (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 ))
133111eqcomd 2616 . . . . . . . . . . . . . 14 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → 𝑙 = ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗))
134133eqeq1d 2612 . . . . . . . . . . . . 13 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (𝑙 = 𝑌 ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))
135132, 134imbi12d 333 . . . . . . . . . . . 12 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ ((𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))) ∧ 𝑦 finSupp 𝑌)) → (((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌) ↔ ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)))
136135anassrs 678 . . . . . . . . . . 11 (((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) ∧ 𝑦 finSupp 𝑌) → (((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌) ↔ ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)))
137136pm5.74da 719 . . . . . . . . . 10 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → ((𝑦 finSupp 𝑌 → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌)) ↔ (𝑦 finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
138 impexp 461 . . . . . . . . . . 11 (((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ (𝑦 finSupp 𝑌 → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌)))
139138a1i 11 . . . . . . . . . 10 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ (𝑦 finSupp 𝑌 → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))) → 𝑙 = 𝑌))))
14064bicomd 212 . . . . . . . . . . 11 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌𝑦 finSupp 𝑌))
141140imbi1d 330 . . . . . . . . . 10 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)) ↔ (𝑦 finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
142137, 139, 1413bitr4d 299 . . . . . . . . 9 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ (𝑙 ∈ (Base‘𝑅) ∧ 𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗})))) → (((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
1431422ralbidva 2971 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
144 breq1 4586 . . . . . . . . . . 11 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥 finSupp 𝑌 ↔ (𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌))
145 oveq1 6556 . . . . . . . . . . . . . 14 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥𝑓 · 𝐹) = ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹))
146145oveq2d 6565 . . . . . . . . . . . . 13 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑊 Σg (𝑥𝑓 · 𝐹)) = (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)))
147146eqeq1d 2612 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 ↔ (𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 ))
148 fveq1 6102 . . . . . . . . . . . . 13 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (𝑥𝑗) = ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗))
149148eqeq1d 2612 . . . . . . . . . . . 12 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑥𝑗) = 𝑌 ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))
150147, 149imbi12d 333 . . . . . . . . . . 11 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → (((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌) ↔ ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌)))
151144, 150imbi12d 333 . . . . . . . . . 10 (𝑥 = (𝑦 ∪ {⟨𝑗, 𝑙⟩}) → ((𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)) ↔ ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
152151ralxpmap 7793 . . . . . . . . 9 (𝑗𝐼 → (∀𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
153152adantl 481 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 ∪ {⟨𝑗, 𝑙⟩}) finSupp 𝑌 → ((𝑊 Σg ((𝑦 ∪ {⟨𝑗, 𝑙⟩}) ∘𝑓 · 𝐹)) = 0 → ((𝑦 ∪ {⟨𝑗, 𝑙⟩})‘𝑗) = 𝑌))))
154143, 153bitr4d 270 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ∀𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌))))
155 breq1 4586 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧 finSupp 𝑌𝑥 finSupp 𝑌))
156155ralrab 3335 . . . . . . 7 (∀𝑥 ∈ {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌) ↔ ∀𝑥 ∈ ((Base‘𝑅) ↑𝑚 𝐼)(𝑥 finSupp 𝑌 → ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
157154, 156syl6bbr 277 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ ∀𝑥 ∈ {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
158 resima 5351 . . . . . . . . . . . . 13 ((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗})) = (𝐹 “ (𝐼 ∖ {𝑗}))
159158eqcomi 2619 . . . . . . . . . . . 12 (𝐹 “ (𝐼 ∖ {𝑗})) = ((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))
160159fveq2i 6106 . . . . . . . . . . 11 ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) = ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗})))
161160eleq2i 2680 . . . . . . . . . 10 ((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))))
162 eqid 2610 . . . . . . . . . . 11 (LSpan‘𝑊) = (LSpan‘𝑊)
16379, 30, 31sylancl 693 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (𝐹 ↾ (𝐼 ∖ {𝑗})):(𝐼 ∖ {𝑗})⟶𝐵)
164 simpl1 1057 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝑊 ∈ LMod)
165243ad2ant2 1076 . . . . . . . . . . . 12 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐼 ∖ {𝑗}) ∈ V)
166165adantr 480 . . . . . . . . . . 11 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (𝐼 ∖ {𝑗}) ∈ V)
167162, 7, 12, 8, 34, 9, 163, 164, 166ellspd 19960 . . . . . . . . . 10 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘((𝐹 ↾ (𝐼 ∖ {𝑗})) “ (𝐼 ∖ {𝑗}))) ↔ ∃𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))))))
168161, 167syl5bb 271 . . . . . . . . 9 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → ((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∃𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗})))))))
169168imbi1d 330 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ (∃𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌)))
170 r19.23v 3005 . . . . . . . 8 (∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌) ↔ (∃𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))(𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌))
171169, 170syl6bbr 277 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ ∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌)))
172171ralbidv 2969 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ (Base‘𝑅)((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ ∀𝑙 ∈ (Base‘𝑅)∀𝑦 ∈ ((Base‘𝑅) ↑𝑚 (𝐼 ∖ {𝑗}))((𝑦 finSupp 𝑌 ∧ (((invg𝑅)‘𝑙) · (𝐹𝑗)) = (𝑊 Σg (𝑦𝑓 · (𝐹 ↾ (𝐼 ∖ {𝑗}))))) → 𝑙 = 𝑌)))
173 fvex 6113 . . . . . . . . . . . 12 (Scalar‘𝑊) ∈ V
1748, 173eqeltri 2684 . . . . . . . . . . 11 𝑅 ∈ V
175 eqid 2610 . . . . . . . . . . . 12 (𝑅 freeLMod 𝐼) = (𝑅 freeLMod 𝐼)
176 eqid 2610 . . . . . . . . . . . 12 {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌}
177175, 12, 34, 176frlmbas 19918 . . . . . . . . . . 11 ((𝑅 ∈ V ∧ 𝐼𝑋) → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
178174, 177mpan 702 . . . . . . . . . 10 (𝐼𝑋 → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
1791783ad2ant2 1076 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
180179adantr 480 . . . . . . . 8 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} = (Base‘(𝑅 freeLMod 𝐼)))
181 islindf4.l . . . . . . . 8 𝐿 = (Base‘(𝑅 freeLMod 𝐼))
182180, 181syl6reqr 2663 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → 𝐿 = {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌})
183182raleqdv 3121 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌) ↔ ∀𝑥 ∈ {𝑧 ∈ ((Base‘𝑅) ↑𝑚 𝐼) ∣ 𝑧 finSupp 𝑌} ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
184157, 172, 1833bitr4d 299 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ (Base‘𝑅)((((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) → 𝑙 = 𝑌) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
1851, 184syl5bb 271 . . . 4 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
1868lmodfgrp 18695 . . . . . . . 8 (𝑊 ∈ LMod → 𝑅 ∈ Grp)
18712, 34, 11grpinvnzcl 17310 . . . . . . . 8 ((𝑅 ∈ Grp ∧ 𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑙) ∈ ((Base‘𝑅) ∖ {𝑌}))
188186, 187sylan 487 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑙) ∈ ((Base‘𝑅) ∖ {𝑌}))
18912, 34, 11grpinvnzcl 17310 . . . . . . . . 9 ((𝑅 ∈ Grp ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}))
190186, 189sylan 487 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}))
191 eldifi 3694 . . . . . . . . . 10 (𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) → 𝑘 ∈ (Base‘𝑅))
19212, 11grpinvinv 17305 . . . . . . . . . 10 ((𝑅 ∈ Grp ∧ 𝑘 ∈ (Base‘𝑅)) → ((invg𝑅)‘((invg𝑅)‘𝑘)) = 𝑘)
193186, 191, 192syl2an 493 . . . . . . . . 9 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ((invg𝑅)‘((invg𝑅)‘𝑘)) = 𝑘)
194193eqcomd 2616 . . . . . . . 8 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → 𝑘 = ((invg𝑅)‘((invg𝑅)‘𝑘)))
195 fveq2 6103 . . . . . . . . . 10 (𝑙 = ((invg𝑅)‘𝑘) → ((invg𝑅)‘𝑙) = ((invg𝑅)‘((invg𝑅)‘𝑘)))
196195eqeq2d 2620 . . . . . . . . 9 (𝑙 = ((invg𝑅)‘𝑘) → (𝑘 = ((invg𝑅)‘𝑙) ↔ 𝑘 = ((invg𝑅)‘((invg𝑅)‘𝑘))))
197196rspcev 3282 . . . . . . . 8 ((((invg𝑅)‘𝑘) ∈ ((Base‘𝑅) ∖ {𝑌}) ∧ 𝑘 = ((invg𝑅)‘((invg𝑅)‘𝑘))) → ∃𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})𝑘 = ((invg𝑅)‘𝑙))
198190, 194, 197syl2anc 691 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑘 ∈ ((Base‘𝑅) ∖ {𝑌})) → ∃𝑙 ∈ ((Base‘𝑅) ∖ {𝑌})𝑘 = ((invg𝑅)‘𝑙))
199 oveq1 6556 . . . . . . . . . 10 (𝑘 = ((invg𝑅)‘𝑙) → (𝑘 · (𝐹𝑗)) = (((invg𝑅)‘𝑙) · (𝐹𝑗)))
200199eleq1d 2672 . . . . . . . . 9 (𝑘 = ((invg𝑅)‘𝑙) → ((𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
201200notbid 307 . . . . . . . 8 (𝑘 = ((invg𝑅)‘𝑙) → (¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
202201adantl 481 . . . . . . 7 ((𝑊 ∈ LMod ∧ 𝑘 = ((invg𝑅)‘𝑙)) → (¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
203188, 198, 202ralxfrd 4805 . . . . . 6 (𝑊 ∈ LMod → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
2042033ad2ant1 1075 . . . . 5 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
205204adantr 480 . . . 4 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑙 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (((invg𝑅)‘𝑙) · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
206 simplr 788 . . . . . . . 8 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → 𝑗𝐼)
207 fvex 6113 . . . . . . . . . 10 (0g𝑅) ∈ V
20834, 207eqeltri 2684 . . . . . . . . 9 𝑌 ∈ V
209208fvconst2 6374 . . . . . . . 8 (𝑗𝐼 → ((𝐼 × {𝑌})‘𝑗) = 𝑌)
210206, 209syl 17 . . . . . . 7 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → ((𝐼 × {𝑌})‘𝑗) = 𝑌)
211210eqeq2d 2620 . . . . . 6 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → ((𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗) ↔ (𝑥𝑗) = 𝑌))
212211imbi2d 329 . . . . 5 ((((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) ∧ 𝑥𝐿) → (((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
213212ralbidva 2968 . . . 4 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = 𝑌)))
214185, 205, 2133bitr4d 299 . . 3 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑗𝐼) → (∀𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
215214ralbidva 2968 . 2 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (∀𝑗𝐼𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗}))) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
2167, 9, 162, 8, 12, 34islindf2 19972 . 2 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑗𝐼𝑘 ∈ ((Base‘𝑅) ∖ {𝑌}) ¬ (𝑘 · (𝐹𝑗)) ∈ ((LSpan‘𝑊)‘(𝐹 “ (𝐼 ∖ {𝑗})))))
217175, 12, 181frlmbasf 19923 . . . . . . . 8 ((𝐼𝑋𝑥𝐿) → 𝑥:𝐼⟶(Base‘𝑅))
2182173ad2antl2 1217 . . . . . . 7 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → 𝑥:𝐼⟶(Base‘𝑅))
219 ffn 5958 . . . . . . 7 (𝑥:𝐼⟶(Base‘𝑅) → 𝑥 Fn 𝐼)
220218, 219syl 17 . . . . . 6 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → 𝑥 Fn 𝐼)
221 fnconstg 6006 . . . . . . 7 (𝑌 ∈ V → (𝐼 × {𝑌}) Fn 𝐼)
222208, 221ax-mp 5 . . . . . 6 (𝐼 × {𝑌}) Fn 𝐼
223 eqfnfv 6219 . . . . . 6 ((𝑥 Fn 𝐼 ∧ (𝐼 × {𝑌}) Fn 𝐼) → (𝑥 = (𝐼 × {𝑌}) ↔ ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
224220, 222, 223sylancl 693 . . . . 5 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → (𝑥 = (𝐼 × {𝑌}) ↔ ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
225224imbi2d 329 . . . 4 (((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) ∧ 𝑥𝐿) → (((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0𝑥 = (𝐼 × {𝑌})) ↔ ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
226225ralbidva 2968 . . 3 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0𝑥 = (𝐼 × {𝑌})) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
227 r19.21v 2943 . . . . 5 (∀𝑗𝐼 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
228227ralbii 2963 . . . 4 (∀𝑥𝐿𝑗𝐼 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
229 ralcom 3079 . . . 4 (∀𝑥𝐿𝑗𝐼 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
230228, 229bitr3i 265 . . 3 (∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → ∀𝑗𝐼 (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗)))
231226, 230syl6bb 275 . 2 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0𝑥 = (𝐼 × {𝑌})) ↔ ∀𝑗𝐼𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0 → (𝑥𝑗) = ((𝐼 × {𝑌})‘𝑗))))
232215, 216, 2313bitr4d 299 1 ((𝑊 ∈ LMod ∧ 𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥𝐿 ((𝑊 Σg (𝑥𝑓 · 𝐹)) = 0𝑥 = (𝐼 × {𝑌}))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   × cxp 5036  dom cdm 5038   ↾ cres 5040   “ cima 5041  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ∘𝑓 cof 6793   ↑𝑚 cmap 7744   finSupp cfsupp 8158  Basecbs 15695  +gcplusg 15768  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923   Σg cgsu 15924  Mndcmnd 17117  Grpcgrp 17245  invgcminusg 17246  CMndccmn 18016  LModclmod 18686  LSpanclspn 18792   freeLMod cfrlm 19909   LIndF clindf 19962 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  df-supp 7183  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fsupp 8159  df-sup 8231  df-oi 8298  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-dec 11370  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-mulr 15782  df-sca 15784  df-vsca 15785  df-ip 15786  df-tset 15787  df-ple 15788  df-ds 15791  df-hom 15793  df-cco 15794  df-0g 15925  df-gsum 15926  df-prds 15931  df-pws 15933  df-mre 16069  df-mrc 16070  df-acs 16072  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-mhm 17158  df-submnd 17159  df-grp 17248  df-minusg 17249  df-sbg 17250  df-mulg 17364  df-subg 17414  df-ghm 17481  df-cntz 17573  df-cmn 18018  df-abl 18019  df-mgp 18313  df-ur 18325  df-ring 18372  df-subrg 18601  df-lmod 18688  df-lss 18754  df-lsp 18793  df-lmhm 18843  df-lbs 18896  df-sra 18993  df-rgmod 18994  df-nzr 19079  df-dsmm 19895  df-frlm 19910  df-uvc 19941  df-lindf 19964 This theorem is referenced by:  islindf5  19997  matunitlindflem1  32575  aacllem  42356
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