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Theorem lindfmm 19985
 Description: Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
Hypotheses
Ref Expression
lindfmm.b 𝐵 = (Base‘𝑆)
lindfmm.c 𝐶 = (Base‘𝑇)
Assertion
Ref Expression
lindfmm ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))

Proof of Theorem lindfmm
Dummy variables 𝑘 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rellindf 19966 . . . . 5 Rel LIndF
21brrelexi 5082 . . . 4 (𝐹 LIndF 𝑆𝐹 ∈ V)
3 simp3 1056 . . . 4 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → 𝐹:𝐼𝐵)
4 dmfex 7017 . . . 4 ((𝐹 ∈ V ∧ 𝐹:𝐼𝐵) → 𝐼 ∈ V)
52, 3, 4syl2anr 494 . . 3 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) ∧ 𝐹 LIndF 𝑆) → 𝐼 ∈ V)
65ex 449 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆𝐼 ∈ V))
71brrelexi 5082 . . . 4 ((𝐺𝐹) LIndF 𝑇 → (𝐺𝐹) ∈ V)
8 f1f 6014 . . . . . 6 (𝐺:𝐵1-1𝐶𝐺:𝐵𝐶)
9 fco 5971 . . . . . 6 ((𝐺:𝐵𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
108, 9sylan 487 . . . . 5 ((𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
11103adant1 1072 . . . 4 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐺𝐹):𝐼𝐶)
12 dmfex 7017 . . . 4 (((𝐺𝐹) ∈ V ∧ (𝐺𝐹):𝐼𝐶) → 𝐼 ∈ V)
137, 11, 12syl2anr 494 . . 3 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) ∧ (𝐺𝐹) LIndF 𝑇) → 𝐼 ∈ V)
1413ex 449 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → ((𝐺𝐹) LIndF 𝑇𝐼 ∈ V))
15 eldifi 3694 . . . . . . . . 9 (𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
16 simpllr 795 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺:𝐵1-1𝐶)
17 lmhmlmod1 18854 . . . . . . . . . . . . . . 15 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)
1817ad3antrrr 762 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑆 ∈ LMod)
19 simprr 792 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝑘 ∈ (Base‘(Scalar‘𝑆)))
20 simprl 790 . . . . . . . . . . . . . . 15 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐹:𝐼𝐵)
21 simpl 472 . . . . . . . . . . . . . . 15 ((𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆))) → 𝑥𝐼)
22 ffvelrn 6265 . . . . . . . . . . . . . . 15 ((𝐹:𝐼𝐵𝑥𝐼) → (𝐹𝑥) ∈ 𝐵)
2320, 21, 22syl2an 493 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐹𝑥) ∈ 𝐵)
24 lindfmm.b . . . . . . . . . . . . . . 15 𝐵 = (Base‘𝑆)
25 eqid 2610 . . . . . . . . . . . . . . 15 (Scalar‘𝑆) = (Scalar‘𝑆)
26 eqid 2610 . . . . . . . . . . . . . . 15 ( ·𝑠𝑆) = ( ·𝑠𝑆)
27 eqid 2610 . . . . . . . . . . . . . . 15 (Base‘(Scalar‘𝑆)) = (Base‘(Scalar‘𝑆))
2824, 25, 26, 27lmodvscl 18703 . . . . . . . . . . . . . 14 ((𝑆 ∈ LMod ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑥) ∈ 𝐵) → (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵)
2918, 19, 23, 28syl3anc 1318 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵)
30 imassrn 5396 . . . . . . . . . . . . . . . 16 (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ ran 𝐹
31 frn 5966 . . . . . . . . . . . . . . . . 17 (𝐹:𝐼𝐵 → ran 𝐹𝐵)
3231adantr 480 . . . . . . . . . . . . . . . 16 ((𝐹:𝐼𝐵𝐼 ∈ V) → ran 𝐹𝐵)
3330, 32syl5ss 3579 . . . . . . . . . . . . . . 15 ((𝐹:𝐼𝐵𝐼 ∈ V) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
3433ad2antlr 759 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵)
35 eqid 2610 . . . . . . . . . . . . . . 15 (LSpan‘𝑆) = (LSpan‘𝑆)
3624, 35lspssv 18804 . . . . . . . . . . . . . 14 ((𝑆 ∈ LMod ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
3718, 34, 36syl2anc 691 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵)
38 f1elima 6421 . . . . . . . . . . . . 13 ((𝐺:𝐵1-1𝐶 ∧ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ 𝐵 ∧ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ⊆ 𝐵) → ((𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) ∈ (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) ↔ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
3916, 29, 37, 38syl3anc 1318 . . . . . . . . . . . 12 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) ∈ (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) ↔ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
40 simplll 794 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → 𝐺 ∈ (𝑆 LMHom 𝑇))
41 eqid 2610 . . . . . . . . . . . . . . . 16 ( ·𝑠𝑇) = ( ·𝑠𝑇)
4225, 27, 24, 26, 41lmhmlin 18856 . . . . . . . . . . . . . . 15 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆)) ∧ (𝐹𝑥) ∈ 𝐵) → (𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) = (𝑘( ·𝑠𝑇)(𝐺‘(𝐹𝑥))))
4340, 19, 23, 42syl3anc 1318 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) = (𝑘( ·𝑠𝑇)(𝐺‘(𝐹𝑥))))
44 ffn 5958 . . . . . . . . . . . . . . . . 17 (𝐹:𝐼𝐵𝐹 Fn 𝐼)
4544ad2antrl 760 . . . . . . . . . . . . . . . 16 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐹 Fn 𝐼)
46 fvco2 6183 . . . . . . . . . . . . . . . 16 ((𝐹 Fn 𝐼𝑥𝐼) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4745, 21, 46syl2an 493 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺𝐹)‘𝑥) = (𝐺‘(𝐹𝑥)))
4847oveq2d 6565 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) = (𝑘( ·𝑠𝑇)(𝐺‘(𝐹𝑥))))
4943, 48eqtr4d 2647 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) = (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)))
50 eqid 2610 . . . . . . . . . . . . . . . 16 (LSpan‘𝑇) = (LSpan‘𝑇)
5124, 35, 50lmhmlsp 18870 . . . . . . . . . . . . . . 15 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ (𝐹 “ (𝐼 ∖ {𝑥})) ⊆ 𝐵) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))))
5240, 34, 51syl2anc 691 . . . . . . . . . . . . . 14 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))))
53 imaco 5557 . . . . . . . . . . . . . . 15 ((𝐺𝐹) “ (𝐼 ∖ {𝑥})) = (𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥})))
5453fveq2i 6106 . . . . . . . . . . . . . 14 ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥}))) = ((LSpan‘𝑇)‘(𝐺 “ (𝐹 “ (𝐼 ∖ {𝑥}))))
5552, 54syl6eqr 2662 . . . . . . . . . . . . 13 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) = ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥}))))
5649, 55eleq12d 2682 . . . . . . . . . . . 12 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝐺‘(𝑘( ·𝑠𝑆)(𝐹𝑥))) ∈ (𝐺 “ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))) ↔ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
5739, 56bitr3d 269 . . . . . . . . . . 11 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → ((𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
5857notbid 307 . . . . . . . . . 10 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ (𝑥𝐼𝑘 ∈ (Base‘(Scalar‘𝑆)))) → (¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
5958anassrs 678 . . . . . . . . 9 (((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) ∧ 𝑘 ∈ (Base‘(Scalar‘𝑆))) → (¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
6015, 59sylan2 490 . . . . . . . 8 (((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) ∧ 𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))})) → (¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
6160ralbidva 2968 . . . . . . 7 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
62 eqid 2610 . . . . . . . . . . . 12 (Scalar‘𝑇) = (Scalar‘𝑇)
6325, 62lmhmsca 18851 . . . . . . . . . . 11 (𝐺 ∈ (𝑆 LMHom 𝑇) → (Scalar‘𝑇) = (Scalar‘𝑆))
6463fveq2d 6107 . . . . . . . . . 10 (𝐺 ∈ (𝑆 LMHom 𝑇) → (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑆)))
6563fveq2d 6107 . . . . . . . . . . 11 (𝐺 ∈ (𝑆 LMHom 𝑇) → (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑆)))
6665sneqd 4137 . . . . . . . . . 10 (𝐺 ∈ (𝑆 LMHom 𝑇) → {(0g‘(Scalar‘𝑇))} = {(0g‘(Scalar‘𝑆))})
6764, 66difeq12d 3691 . . . . . . . . 9 (𝐺 ∈ (𝑆 LMHom 𝑇) → ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}))
6867ad3antrrr 762 . . . . . . . 8 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) → ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) = ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}))
6968raleqdv 3121 . . . . . . 7 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
7061, 69bitr4d 270 . . . . . 6 ((((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) ∧ 𝑥𝐼) → (∀𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
7170ralbidva 2968 . . . . 5 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥}))) ↔ ∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
7217ad2antrr 758 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝑆 ∈ LMod)
73 simprr 792 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝐼 ∈ V)
74 eqid 2610 . . . . . . 7 (0g‘(Scalar‘𝑆)) = (0g‘(Scalar‘𝑆))
7524, 26, 35, 25, 27, 74islindf2 19972 . . . . . 6 ((𝑆 ∈ LMod ∧ 𝐼 ∈ V ∧ 𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆 ↔ ∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
7672, 73, 20, 75syl3anc 1318 . . . . 5 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐹 LIndF 𝑆 ↔ ∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑆)) ∖ {(0g‘(Scalar‘𝑆))}) ¬ (𝑘( ·𝑠𝑆)(𝐹𝑥)) ∈ ((LSpan‘𝑆)‘(𝐹 “ (𝐼 ∖ {𝑥})))))
77 lmhmlmod2 18853 . . . . . . 7 (𝐺 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)
7877ad2antrr 758 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → 𝑇 ∈ LMod)
7910ad2ant2lr 780 . . . . . 6 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐺𝐹):𝐼𝐶)
80 lindfmm.c . . . . . . 7 𝐶 = (Base‘𝑇)
81 eqid 2610 . . . . . . 7 (Base‘(Scalar‘𝑇)) = (Base‘(Scalar‘𝑇))
82 eqid 2610 . . . . . . 7 (0g‘(Scalar‘𝑇)) = (0g‘(Scalar‘𝑇))
8380, 41, 50, 62, 81, 82islindf2 19972 . . . . . 6 ((𝑇 ∈ LMod ∧ 𝐼 ∈ V ∧ (𝐺𝐹):𝐼𝐶) → ((𝐺𝐹) LIndF 𝑇 ↔ ∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
8478, 73, 79, 83syl3anc 1318 . . . . 5 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → ((𝐺𝐹) LIndF 𝑇 ↔ ∀𝑥𝐼𝑘 ∈ ((Base‘(Scalar‘𝑇)) ∖ {(0g‘(Scalar‘𝑇))}) ¬ (𝑘( ·𝑠𝑇)((𝐺𝐹)‘𝑥)) ∈ ((LSpan‘𝑇)‘((𝐺𝐹) “ (𝐼 ∖ {𝑥})))))
8571, 76, 843bitr4d 299 . . . 4 (((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) ∧ (𝐹:𝐼𝐵𝐼 ∈ V)) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))
8685exp32 629 . . 3 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶) → (𝐹:𝐼𝐵 → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))))
87863impia 1253 . 2 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐼 ∈ V → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇)))
886, 14, 87pm5.21ndd 368 1 ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  Vcvv 3173   ∖ cdif 3537   ⊆ wss 3540  {csn 4125   class class class wbr 4583  ran crn 5039   “ cima 5041   ∘ ccom 5042   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  Scalarcsca 15771   ·𝑠 cvsca 15772  0gc0g 15923  LModclmod 18686  LSpanclspn 18792   LMHom clmhm 18840   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-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-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-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-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  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-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-sbg 17250  df-subg 17414  df-ghm 17481  df-mgp 18313  df-ur 18325  df-ring 18372  df-lmod 18688  df-lss 18754  df-lsp 18793  df-lmhm 18843  df-lindf 19964 This theorem is referenced by:  lindsmm  19986
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