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Theorem pwssplit4 36677
 Description: Splitting for structure powers 4: maps isomorphically onto the other half. (Contributed by Stefan O'Rear, 25-Jan-2015.)
Hypotheses
Ref Expression
pwssplit4.e 𝐸 = (𝑅s (𝐴𝐵))
pwssplit4.g 𝐺 = (Base‘𝐸)
pwssplit4.z 0 = (0g𝑅)
pwssplit4.k 𝐾 = {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })}
pwssplit4.f 𝐹 = (𝑥𝐾 ↦ (𝑥𝐵))
pwssplit4.c 𝐶 = (𝑅s 𝐴)
pwssplit4.d 𝐷 = (𝑅s 𝐵)
pwssplit4.l 𝐿 = (𝐸s 𝐾)
Assertion
Ref Expression
pwssplit4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦   𝑥,𝐷,𝑦   𝑥,𝐸,𝑦   𝑥,𝐺,𝑦   𝑥,𝐾   𝑥,𝐿   𝑥,𝑅,𝑦   𝑥,𝑉,𝑦   𝑥, 0 ,𝑦
Allowed substitution hints:   𝐹(𝑥,𝑦)   𝐾(𝑦)   𝐿(𝑦)

Proof of Theorem pwssplit4
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 pwssplit4.f . . . 4 𝐹 = (𝑥𝐾 ↦ (𝑥𝐵))
2 pwssplit4.k . . . . . 6 𝐾 = {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })}
3 ssrab2 3650 . . . . . 6 {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })} ⊆ 𝐺
42, 3eqsstri 3598 . . . . 5 𝐾𝐺
5 resmpt 5369 . . . . 5 (𝐾𝐺 → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) = (𝑥𝐾 ↦ (𝑥𝐵)))
64, 5ax-mp 5 . . . 4 ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) = (𝑥𝐾 ↦ (𝑥𝐵))
71, 6eqtr4i 2635 . . 3 𝐹 = ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾)
8 ssun2 3739 . . . . . 6 𝐵 ⊆ (𝐴𝐵)
98a1i 11 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐵 ⊆ (𝐴𝐵))
10 pwssplit4.e . . . . . 6 𝐸 = (𝑅s (𝐴𝐵))
11 pwssplit4.d . . . . . 6 𝐷 = (𝑅s 𝐵)
12 pwssplit4.g . . . . . 6 𝐺 = (Base‘𝐸)
13 eqid 2610 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
14 eqid 2610 . . . . . 6 (𝑥𝐺 ↦ (𝑥𝐵)) = (𝑥𝐺 ↦ (𝑥𝐵))
1510, 11, 12, 13, 14pwssplit3 18882 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷))
169, 15syld3an3 1363 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷))
17 simp1 1054 . . . . . . . . . 10 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝑅 ∈ LMod)
18 lmodgrp 18693 . . . . . . . . . 10 (𝑅 ∈ LMod → 𝑅 ∈ Grp)
19 grpmnd 17252 . . . . . . . . . 10 (𝑅 ∈ Grp → 𝑅 ∈ Mnd)
2017, 18, 193syl 18 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝑅 ∈ Mnd)
21 ssun1 3738 . . . . . . . . . . 11 𝐴 ⊆ (𝐴𝐵)
22 ssexg 4732 . . . . . . . . . . 11 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐴 ∈ V)
2321, 22mpan 702 . . . . . . . . . 10 ((𝐴𝐵) ∈ 𝑉𝐴 ∈ V)
24233ad2ant2 1076 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐴 ∈ V)
25 pwssplit4.c . . . . . . . . . 10 𝐶 = (𝑅s 𝐴)
26 pwssplit4.z . . . . . . . . . 10 0 = (0g𝑅)
2725, 26pws0g 17149 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × { 0 }) = (0g𝐶))
2820, 24, 27syl2anc 691 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴 × { 0 }) = (0g𝐶))
2928eqeq2d 2620 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ (𝑦𝐴) = (0g𝐶)))
3029rabbidv 3164 . . . . . 6 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })} = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
312, 30syl5eq 2656 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
3221a1i 11 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ (𝐴𝐵))
33 eqid 2610 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
34 eqid 2610 . . . . . . . 8 (𝑦𝐺 ↦ (𝑦𝐴)) = (𝑦𝐺 ↦ (𝑦𝐴))
3510, 25, 12, 33, 34pwssplit3 18882 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉𝐴 ⊆ (𝐴𝐵)) → (𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶))
3632, 35syld3an3 1363 . . . . . 6 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶))
37 fvex 6113 . . . . . . . . 9 (0g𝐶) ∈ V
3834mptiniseg 5546 . . . . . . . . 9 ((0g𝐶) ∈ V → ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)}) = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
3937, 38ax-mp 5 . . . . . . . 8 ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)}) = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)}
4039eqcomi 2619 . . . . . . 7 {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)} = ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)})
41 eqid 2610 . . . . . . 7 (0g𝐶) = (0g𝐶)
42 eqid 2610 . . . . . . 7 (LSubSp‘𝐸) = (LSubSp‘𝐸)
4340, 41, 42lmhmkerlss 18872 . . . . . 6 ((𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶) → {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)} ∈ (LSubSp‘𝐸))
4436, 43syl 17 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)} ∈ (LSubSp‘𝐸))
4531, 44eqeltrd 2688 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 ∈ (LSubSp‘𝐸))
46 pwssplit4.l . . . . 5 𝐿 = (𝐸s 𝐾)
4742, 46reslmhm 18873 . . . 4 (((𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷) ∧ 𝐾 ∈ (LSubSp‘𝐸)) → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
4816, 45, 47syl2anc 691 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
497, 48syl5eqel 2692 . 2 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMHom 𝐷))
501fvtresfn 6193 . . . . . . 7 (𝑎𝐾 → (𝐹𝑎) = (𝑎𝐵))
51 ssexg 4732 . . . . . . . . . . 11 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐵 ∈ V)
528, 51mpan 702 . . . . . . . . . 10 ((𝐴𝐵) ∈ 𝑉𝐵 ∈ V)
53523ad2ant2 1076 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐵 ∈ V)
5411, 26pws0g 17149 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐵 ∈ V) → (𝐵 × { 0 }) = (0g𝐷))
5520, 53, 54syl2anc 691 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐵 × { 0 }) = (0g𝐷))
5655eqcomd 2616 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (0g𝐷) = (𝐵 × { 0 }))
5750, 56eqeqan12rd 2628 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝐹𝑎) = (0g𝐷) ↔ (𝑎𝐵) = (𝐵 × { 0 })))
58 reseq1 5311 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝑦𝐴) = (𝑎𝐴))
5958eqeq1d 2612 . . . . . . . . 9 (𝑦 = 𝑎 → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ (𝑎𝐴) = (𝐴 × { 0 })))
6059, 2elrab2 3333 . . . . . . . 8 (𝑎𝐾 ↔ (𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })))
61 uneq12 3724 . . . . . . . . . . . . 13 (((𝑎𝐴) = (𝐴 × { 0 }) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → ((𝑎𝐴) ∪ (𝑎𝐵)) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 })))
62 resundi 5330 . . . . . . . . . . . . 13 (𝑎 ↾ (𝐴𝐵)) = ((𝑎𝐴) ∪ (𝑎𝐵))
63 xpundir 5095 . . . . . . . . . . . . 13 ((𝐴𝐵) × { 0 }) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 }))
6461, 62, 633eqtr4g 2669 . . . . . . . . . . . 12 (((𝑎𝐴) = (𝐴 × { 0 }) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
6564adantll 746 . . . . . . . . . . 11 (((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
6665adantl 481 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
67 eqid 2610 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
68 simpl1 1057 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑅 ∈ LMod)
69 simp2 1055 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ 𝑉)
7069adantr 480 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → (𝐴𝐵) ∈ 𝑉)
71 simprll 798 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎𝐺)
7210, 67, 12, 68, 70, 71pwselbas 15972 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎:(𝐴𝐵)⟶(Base‘𝑅))
73 ffn 5958 . . . . . . . . . . 11 (𝑎:(𝐴𝐵)⟶(Base‘𝑅) → 𝑎 Fn (𝐴𝐵))
74 fnresdm 5914 . . . . . . . . . . 11 (𝑎 Fn (𝐴𝐵) → (𝑎 ↾ (𝐴𝐵)) = 𝑎)
7572, 73, 743syl 18 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → (𝑎 ↾ (𝐴𝐵)) = 𝑎)
7610, 26pws0g 17149 . . . . . . . . . . . . 13 ((𝑅 ∈ Mnd ∧ (𝐴𝐵) ∈ 𝑉) → ((𝐴𝐵) × { 0 }) = (0g𝐸))
7720, 69, 76syl2anc 691 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) × { 0 }) = (0g𝐸))
7810pwslmod 18791 . . . . . . . . . . . . . . 15 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉) → 𝐸 ∈ LMod)
79783adant3 1074 . . . . . . . . . . . . . 14 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐸 ∈ LMod)
8042lsssubg 18778 . . . . . . . . . . . . . 14 ((𝐸 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝐸)) → 𝐾 ∈ (SubGrp‘𝐸))
8179, 45, 80syl2anc 691 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 ∈ (SubGrp‘𝐸))
82 eqid 2610 . . . . . . . . . . . . . 14 (0g𝐸) = (0g𝐸)
8346, 82subg0 17423 . . . . . . . . . . . . 13 (𝐾 ∈ (SubGrp‘𝐸) → (0g𝐸) = (0g𝐿))
8481, 83syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (0g𝐸) = (0g𝐿))
8577, 84eqtrd 2644 . . . . . . . . . . 11 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) × { 0 }) = (0g𝐿))
8685adantr 480 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → ((𝐴𝐵) × { 0 }) = (0g𝐿))
8766, 75, 863eqtr3d 2652 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎 = (0g𝐿))
8887exp32 629 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿))))
8960, 88syl5bi 231 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑎𝐾 → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿))))
9089imp 444 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿)))
9157, 90sylbid 229 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿)))
9291ralrimiva 2949 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ∀𝑎𝐾 ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿)))
93 lmghm 18852 . . . . 5 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹 ∈ (𝐿 GrpHom 𝐷))
9446, 12ressbas2 15758 . . . . . . 7 (𝐾𝐺𝐾 = (Base‘𝐿))
954, 94ax-mp 5 . . . . . 6 𝐾 = (Base‘𝐿)
96 eqid 2610 . . . . . 6 (0g𝐿) = (0g𝐿)
97 eqid 2610 . . . . . 6 (0g𝐷) = (0g𝐷)
9895, 13, 96, 97ghmf1 17512 . . . . 5 (𝐹 ∈ (𝐿 GrpHom 𝐷) → (𝐹:𝐾1-1→(Base‘𝐷) ↔ ∀𝑎𝐾 ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿))))
9949, 93, 983syl 18 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐹:𝐾1-1→(Base‘𝐷) ↔ ∀𝑎𝐾 ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿))))
10092, 99mpbird 246 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹:𝐾1-1→(Base‘𝐷))
101 eqid 2610 . . . . . 6 (Base‘𝐿) = (Base‘𝐿)
102101, 13lmhmf 18855 . . . . 5 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:(Base‘𝐿)⟶(Base‘𝐷))
103 frn 5966 . . . . 5 (𝐹:(Base‘𝐿)⟶(Base‘𝐷) → ran 𝐹 ⊆ (Base‘𝐷))
10449, 102, 1033syl 18 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ran 𝐹 ⊆ (Base‘𝐷))
10511, 67, 13pwselbasb 15971 . . . . . . . . . . . . . . 15 ((𝑅 ∈ LMod ∧ 𝐵 ∈ V) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
10617, 53, 105syl2anc 691 . . . . . . . . . . . . . 14 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
107106biimpa 500 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎:𝐵⟶(Base‘𝑅))
108 fvex 6113 . . . . . . . . . . . . . . . . 17 (0g𝑅) ∈ V
10926, 108eqeltri 2684 . . . . . . . . . . . . . . . 16 0 ∈ V
110109fconst 6004 . . . . . . . . . . . . . . 15 (𝐴 × { 0 }):𝐴⟶{ 0 }
111110a1i 11 . . . . . . . . . . . . . 14 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶{ 0 })
11220adantr 480 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑅 ∈ Mnd)
11367, 26mndidcl 17131 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅))
114112, 113syl 17 . . . . . . . . . . . . . . 15 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 0 ∈ (Base‘𝑅))
115114snssd 4281 . . . . . . . . . . . . . 14 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → { 0 } ⊆ (Base‘𝑅))
116111, 115fssd 5970 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶(Base‘𝑅))
117 incom 3767 . . . . . . . . . . . . . . 15 (𝐵𝐴) = (𝐴𝐵)
118 simp3 1056 . . . . . . . . . . . . . . 15 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
119117, 118syl5eq 2656 . . . . . . . . . . . . . 14 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐵𝐴) = ∅)
120119adantr 480 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵𝐴) = ∅)
121 fun 5979 . . . . . . . . . . . . 13 (((𝑎:𝐵⟶(Base‘𝑅) ∧ (𝐴 × { 0 }):𝐴⟶(Base‘𝑅)) ∧ (𝐵𝐴) = ∅) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
122107, 116, 120, 121syl21anc 1317 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
123 uncom 3719 . . . . . . . . . . . . 13 (𝐵𝐴) = (𝐴𝐵)
124 unidm 3718 . . . . . . . . . . . . 13 ((Base‘𝑅) ∪ (Base‘𝑅)) = (Base‘𝑅)
125123, 124feq23i 5952 . . . . . . . . . . . 12 ((𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)) ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅))
126122, 125sylib 207 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅))
12710, 67, 12pwselbasb 15971 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
1281273adant3 1074 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
129128adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
130126, 129mpbird 246 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺)
131 simpl3 1059 . . . . . . . . . . . . . 14 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴𝐵) = ∅)
132117, 131syl5eq 2656 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵𝐴) = ∅)
133 ffn 5958 . . . . . . . . . . . . . 14 (𝑎:𝐵⟶(Base‘𝑅) → 𝑎 Fn 𝐵)
134 fnresdisj 5915 . . . . . . . . . . . . . 14 (𝑎 Fn 𝐵 → ((𝐵𝐴) = ∅ ↔ (𝑎𝐴) = ∅))
135107, 133, 1343syl 18 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐵𝐴) = ∅ ↔ (𝑎𝐴) = ∅))
136132, 135mpbid 221 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎𝐴) = ∅)
137 fnconstg 6006 . . . . . . . . . . . . . 14 ( 0 ∈ V → (𝐴 × { 0 }) Fn 𝐴)
138 fnresdm 5914 . . . . . . . . . . . . . 14 ((𝐴 × { 0 }) Fn 𝐴 → ((𝐴 × { 0 }) ↾ 𝐴) = (𝐴 × { 0 }))
139109, 137, 138mp2b 10 . . . . . . . . . . . . 13 ((𝐴 × { 0 }) ↾ 𝐴) = (𝐴 × { 0 })
140139a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐴 × { 0 }) ↾ 𝐴) = (𝐴 × { 0 }))
141136, 140uneq12d 3730 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴)) = (∅ ∪ (𝐴 × { 0 })))
142 resundir 5331 . . . . . . . . . . 11 ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = ((𝑎𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴))
143 uncom 3719 . . . . . . . . . . . 12 (∅ ∪ (𝐴 × { 0 })) = ((𝐴 × { 0 }) ∪ ∅)
144 un0 3919 . . . . . . . . . . . 12 ((𝐴 × { 0 }) ∪ ∅) = (𝐴 × { 0 })
145143, 144eqtr2i 2633 . . . . . . . . . . 11 (𝐴 × { 0 }) = (∅ ∪ (𝐴 × { 0 }))
146141, 142, 1453eqtr4g 2669 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 }))
147 reseq1 5311 . . . . . . . . . . . 12 (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑦𝐴) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴))
148147eqeq1d 2612 . . . . . . . . . . 11 (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
149148, 2elrab2 3333 . . . . . . . . . 10 ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 ↔ ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ∧ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
150130, 146, 149sylanbrc 695 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾)
151 resexg 5362 . . . . . . . . . 10 ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
152150, 151syl 17 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
153 reseq1 5311 . . . . . . . . . 10 (𝑥 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑥𝐵) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
154153, 1fvmptg 6189 . . . . . . . . 9 (((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 ∧ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
155150, 152, 154syl2anc 691 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
156 resundir 5331 . . . . . . . . 9 ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵))
157 fnresdm 5914 . . . . . . . . . . . 12 (𝑎 Fn 𝐵 → (𝑎𝐵) = 𝑎)
158107, 133, 1573syl 18 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎𝐵) = 𝑎)
159 ffn 5958 . . . . . . . . . . . . . . 15 ((𝐴 × { 0 }):𝐴⟶{ 0 } → (𝐴 × { 0 }) Fn 𝐴)
160 fnresdisj 5915 . . . . . . . . . . . . . . 15 ((𝐴 × { 0 }) Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅))
161110, 159, 160mp2b 10 . . . . . . . . . . . . . 14 ((𝐴𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
162161biimpi 205 . . . . . . . . . . . . 13 ((𝐴𝐵) = ∅ → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
1631623ad2ant3 1077 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
164163adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
165158, 164uneq12d 3730 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = (𝑎 ∪ ∅))
166 un0 3919 . . . . . . . . . 10 (𝑎 ∪ ∅) = 𝑎
167165, 166syl6eq 2660 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = 𝑎)
168156, 167syl5eq 2656 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = 𝑎)
169155, 168eqtrd 2644 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = 𝑎)
17095, 13lmhmf 18855 . . . . . . . . . 10 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:𝐾⟶(Base‘𝐷))
171 ffn 5958 . . . . . . . . . 10 (𝐹:𝐾⟶(Base‘𝐷) → 𝐹 Fn 𝐾)
17249, 170, 1713syl 18 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 Fn 𝐾)
173172adantr 480 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝐹 Fn 𝐾)
174 fnfvelrn 6264 . . . . . . . 8 ((𝐹 Fn 𝐾 ∧ (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
175173, 150, 174syl2anc 691 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
176169, 175eqeltrrd 2689 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎 ∈ ran 𝐹)
177176ex 449 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑎 ∈ (Base‘𝐷) → 𝑎 ∈ ran 𝐹))
178177ssrdv 3574 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (Base‘𝐷) ⊆ ran 𝐹)
179104, 178eqssd 3585 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ran 𝐹 = (Base‘𝐷))
180 dff1o5 6059 . . 3 (𝐹:𝐾1-1-onto→(Base‘𝐷) ↔ (𝐹:𝐾1-1→(Base‘𝐷) ∧ ran 𝐹 = (Base‘𝐷)))
181100, 179, 180sylanbrc 695 . 2 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹:𝐾1-1-onto→(Base‘𝐷))
18295, 13islmim 18883 . 2 (𝐹 ∈ (𝐿 LMIso 𝐷) ↔ (𝐹 ∈ (𝐿 LMHom 𝐷) ∧ 𝐹:𝐾1-1-onto→(Base‘𝐷)))
18349, 181, 182sylanbrc 695 1 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125   ↦ cmpt 4643   × cxp 5036  ◡ccnv 5037  ran crn 5039   ↾ cres 5040   “ cima 5041   Fn wfn 5799  ⟶wf 5800  –1-1→wf1 5801  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Basecbs 15695   ↾s cress 15696  0gc0g 15923   ↑s cpws 15930  Mndcmnd 17117  Grpcgrp 17245  SubGrpcsubg 17411   GrpHom cghm 17480  LModclmod 18686  LSubSpclss 18753   LMHom clmhm 18840   LMIso clmim 18841 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-of 6795  df-om 6958  df-1st 7059  df-2nd 7060  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-sup 8231  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-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-prds 15931  df-pws 15933  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-lmhm 18843  df-lmim 18844 This theorem is referenced by:  pwslnmlem2  36681
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