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Theorem pwssplit4 42134
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 4078 . . . . . 6 {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })} ⊆ 𝐺
42, 3eqsstri 4017 . . . . 5 𝐾𝐺
5 resmpt 6038 . . . . 5 (𝐾𝐺 → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) = (𝑥𝐾 ↦ (𝑥𝐵)))
64, 5ax-mp 5 . . . 4 ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) = (𝑥𝐾 ↦ (𝑥𝐵))
71, 6eqtr4i 2762 . . 3 𝐹 = ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾)
8 ssun2 4174 . . . . . 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 2731 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
14 eqid 2731 . . . . . 6 (𝑥𝐺 ↦ (𝑥𝐵)) = (𝑥𝐺 ↦ (𝑥𝐵))
1510, 11, 12, 13, 14pwssplit3 20817 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷))
169, 15syld3an3 1408 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷))
17 simp1 1135 . . . . . . . . . 10 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝑅 ∈ LMod)
18 lmodgrp 20622 . . . . . . . . . 10 (𝑅 ∈ LMod → 𝑅 ∈ Grp)
19 grpmnd 18863 . . . . . . . . . 10 (𝑅 ∈ Grp → 𝑅 ∈ Mnd)
2017, 18, 193syl 18 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝑅 ∈ Mnd)
21 ssun1 4173 . . . . . . . . . . 11 𝐴 ⊆ (𝐴𝐵)
22 ssexg 5324 . . . . . . . . . . 11 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐴 ∈ V)
2321, 22mpan 687 . . . . . . . . . 10 ((𝐴𝐵) ∈ 𝑉𝐴 ∈ V)
24233ad2ant2 1133 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐴 ∈ V)
25 pwssplit4.c . . . . . . . . . 10 𝐶 = (𝑅s 𝐴)
26 pwssplit4.z . . . . . . . . . 10 0 = (0g𝑅)
2725, 26pws0g 18696 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × { 0 }) = (0g𝐶))
2820, 24, 27syl2anc 583 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴 × { 0 }) = (0g𝐶))
2928eqeq2d 2742 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ (𝑦𝐴) = (0g𝐶)))
3029rabbidv 3439 . . . . . 6 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })} = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
312, 30eqtrid 2783 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
3221a1i 11 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ (𝐴𝐵))
33 eqid 2731 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
34 eqid 2731 . . . . . . . 8 (𝑦𝐺 ↦ (𝑦𝐴)) = (𝑦𝐺 ↦ (𝑦𝐴))
3510, 25, 12, 33, 34pwssplit3 20817 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉𝐴 ⊆ (𝐴𝐵)) → (𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶))
3632, 35syld3an3 1408 . . . . . 6 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶))
37 fvex 6905 . . . . . . . . 9 (0g𝐶) ∈ V
3834mptiniseg 6239 . . . . . . . . 9 ((0g𝐶) ∈ V → ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)}) = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
3937, 38ax-mp 5 . . . . . . . 8 ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)}) = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)}
4039eqcomi 2740 . . . . . . 7 {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)} = ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)})
41 eqid 2731 . . . . . . 7 (0g𝐶) = (0g𝐶)
42 eqid 2731 . . . . . . 7 (LSubSp‘𝐸) = (LSubSp‘𝐸)
4340, 41, 42lmhmkerlss 20807 . . . . . 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 2832 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 ∈ (LSubSp‘𝐸))
46 pwssplit4.l . . . . 5 𝐿 = (𝐸s 𝐾)
4742, 46reslmhm 20808 . . . 4 (((𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷) ∧ 𝐾 ∈ (LSubSp‘𝐸)) → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
4816, 45, 47syl2anc 583 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
497, 48eqeltrid 2836 . 2 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMHom 𝐷))
501fvtresfn 7001 . . . . . . 7 (𝑎𝐾 → (𝐹𝑎) = (𝑎𝐵))
51 ssexg 5324 . . . . . . . . . . 11 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐵 ∈ V)
528, 51mpan 687 . . . . . . . . . 10 ((𝐴𝐵) ∈ 𝑉𝐵 ∈ V)
53523ad2ant2 1133 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐵 ∈ V)
5411, 26pws0g 18696 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐵 ∈ V) → (𝐵 × { 0 }) = (0g𝐷))
5520, 53, 54syl2anc 583 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐵 × { 0 }) = (0g𝐷))
5655eqcomd 2737 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (0g𝐷) = (𝐵 × { 0 }))
5750, 56eqeqan12rd 2746 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝐹𝑎) = (0g𝐷) ↔ (𝑎𝐵) = (𝐵 × { 0 })))
58 reseq1 5976 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝑦𝐴) = (𝑎𝐴))
5958eqeq1d 2733 . . . . . . . . 9 (𝑦 = 𝑎 → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ (𝑎𝐴) = (𝐴 × { 0 })))
6059, 2elrab2 3687 . . . . . . . 8 (𝑎𝐾 ↔ (𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })))
61 uneq12 4159 . . . . . . . . . . . . 13 (((𝑎𝐴) = (𝐴 × { 0 }) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → ((𝑎𝐴) ∪ (𝑎𝐵)) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 })))
62 resundi 5996 . . . . . . . . . . . . 13 (𝑎 ↾ (𝐴𝐵)) = ((𝑎𝐴) ∪ (𝑎𝐵))
63 xpundir 5746 . . . . . . . . . . . . 13 ((𝐴𝐵) × { 0 }) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 }))
6461, 62, 633eqtr4g 2796 . . . . . . . . . . . 12 (((𝑎𝐴) = (𝐴 × { 0 }) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
6564adantll 711 . . . . . . . . . . 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 2731 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
68 simpl1 1190 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑅 ∈ LMod)
69 simp2 1136 . . . . . . . . . . . . 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 776 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎𝐺)
7210, 67, 12, 68, 70, 71pwselbas 17440 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎:(𝐴𝐵)⟶(Base‘𝑅))
73 ffn 6718 . . . . . . . . . . 11 (𝑎:(𝐴𝐵)⟶(Base‘𝑅) → 𝑎 Fn (𝐴𝐵))
74 fnresdm 6670 . . . . . . . . . . 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 18696 . . . . . . . . . . . . 13 ((𝑅 ∈ Mnd ∧ (𝐴𝐵) ∈ 𝑉) → ((𝐴𝐵) × { 0 }) = (0g𝐸))
7720, 69, 76syl2anc 583 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) × { 0 }) = (0g𝐸))
7810pwslmod 20726 . . . . . . . . . . . . . . 15 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉) → 𝐸 ∈ LMod)
79783adant3 1131 . . . . . . . . . . . . . 14 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐸 ∈ LMod)
8042lsssubg 20713 . . . . . . . . . . . . . 14 ((𝐸 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝐸)) → 𝐾 ∈ (SubGrp‘𝐸))
8179, 45, 80syl2anc 583 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 ∈ (SubGrp‘𝐸))
82 eqid 2731 . . . . . . . . . . . . . 14 (0g𝐸) = (0g𝐸)
8346, 82subg0 19049 . . . . . . . . . . . . 13 (𝐾 ∈ (SubGrp‘𝐸) → (0g𝐸) = (0g𝐿))
8481, 83syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (0g𝐸) = (0g𝐿))
8577, 84eqtrd 2771 . . . . . . . . . . 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 2779 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎 = (0g𝐿))
8887exp32 420 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿))))
8960, 88biimtrid 241 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑎𝐾 → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿))))
9089imp 406 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿)))
9157, 90sylbid 239 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿)))
9291ralrimiva 3145 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ∀𝑎𝐾 ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿)))
93 lmghm 20787 . . . . 5 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹 ∈ (𝐿 GrpHom 𝐷))
9446, 12ressbas2 17187 . . . . . . 7 (𝐾𝐺𝐾 = (Base‘𝐿))
954, 94ax-mp 5 . . . . . 6 𝐾 = (Base‘𝐿)
96 eqid 2731 . . . . . 6 (0g𝐿) = (0g𝐿)
97 eqid 2731 . . . . . 6 (0g𝐷) = (0g𝐷)
9895, 13, 96, 97ghmf1 19161 . . . . 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 256 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹:𝐾1-1→(Base‘𝐷))
101 eqid 2731 . . . . . 6 (Base‘𝐿) = (Base‘𝐿)
102101, 13lmhmf 20790 . . . . 5 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:(Base‘𝐿)⟶(Base‘𝐷))
103 frn 6725 . . . . 5 (𝐹:(Base‘𝐿)⟶(Base‘𝐷) → ran 𝐹 ⊆ (Base‘𝐷))
10449, 102, 1033syl 18 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ran 𝐹 ⊆ (Base‘𝐷))
105 reseq1 5976 . . . . . . 7 (𝑥 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑥𝐵) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
10611, 67, 13pwselbasb 17439 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ 𝐵 ∈ V) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
10717, 53, 106syl2anc 583 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
108107biimpa 476 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎:𝐵⟶(Base‘𝑅))
10926fvexi 6906 . . . . . . . . . . . . . 14 0 ∈ V
110109fconst 6778 . . . . . . . . . . . . 13 (𝐴 × { 0 }):𝐴⟶{ 0 }
111110a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶{ 0 })
11220adantr 480 . . . . . . . . . . . . . 14 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑅 ∈ Mnd)
11367, 26mndidcl 18675 . . . . . . . . . . . . . 14 (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅))
114112, 113syl 17 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 0 ∈ (Base‘𝑅))
115114snssd 4813 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → { 0 } ⊆ (Base‘𝑅))
116111, 115fssd 6736 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶(Base‘𝑅))
117 incom 4202 . . . . . . . . . . . . 13 (𝐵𝐴) = (𝐴𝐵)
118 simp3 1137 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
119117, 118eqtrid 2783 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐵𝐴) = ∅)
120119adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵𝐴) = ∅)
121 fun 6754 . . . . . . . . . . 11 (((𝑎:𝐵⟶(Base‘𝑅) ∧ (𝐴 × { 0 }):𝐴⟶(Base‘𝑅)) ∧ (𝐵𝐴) = ∅) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
122108, 116, 120, 121syl21anc 835 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
123 uncom 4154 . . . . . . . . . . 11 (𝐵𝐴) = (𝐴𝐵)
124 unidm 4153 . . . . . . . . . . 11 ((Base‘𝑅) ∪ (Base‘𝑅)) = (Base‘𝑅)
125123, 124feq23i 6712 . . . . . . . . . 10 ((𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)) ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅))
126122, 125sylib 217 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅))
12710, 67, 12pwselbasb 17439 . . . . . . . . . . 11 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
1281273adant3 1131 . . . . . . . . . 10 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
129128adantr 480 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
130126, 129mpbird 256 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺)
131 simpl3 1192 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴𝐵) = ∅)
132117, 131eqtrid 2783 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵𝐴) = ∅)
133 ffn 6718 . . . . . . . . . . . 12 (𝑎:𝐵⟶(Base‘𝑅) → 𝑎 Fn 𝐵)
134 fnresdisj 6671 . . . . . . . . . . . 12 (𝑎 Fn 𝐵 → ((𝐵𝐴) = ∅ ↔ (𝑎𝐴) = ∅))
135108, 133, 1343syl 18 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐵𝐴) = ∅ ↔ (𝑎𝐴) = ∅))
136132, 135mpbid 231 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎𝐴) = ∅)
137 fnconstg 6780 . . . . . . . . . . . 12 ( 0 ∈ V → (𝐴 × { 0 }) Fn 𝐴)
138 fnresdm 6670 . . . . . . . . . . . 12 ((𝐴 × { 0 }) Fn 𝐴 → ((𝐴 × { 0 }) ↾ 𝐴) = (𝐴 × { 0 }))
139109, 137, 138mp2b 10 . . . . . . . . . . 11 ((𝐴 × { 0 }) ↾ 𝐴) = (𝐴 × { 0 })
140139a1i 11 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐴 × { 0 }) ↾ 𝐴) = (𝐴 × { 0 }))
141136, 140uneq12d 4165 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴)) = (∅ ∪ (𝐴 × { 0 })))
142 resundir 5997 . . . . . . . . 9 ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = ((𝑎𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴))
143 uncom 4154 . . . . . . . . . 10 (∅ ∪ (𝐴 × { 0 })) = ((𝐴 × { 0 }) ∪ ∅)
144 un0 4391 . . . . . . . . . 10 ((𝐴 × { 0 }) ∪ ∅) = (𝐴 × { 0 })
145143, 144eqtr2i 2760 . . . . . . . . 9 (𝐴 × { 0 }) = (∅ ∪ (𝐴 × { 0 }))
146141, 142, 1453eqtr4g 2796 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 }))
147 reseq1 5976 . . . . . . . . . 10 (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑦𝐴) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴))
148147eqeq1d 2733 . . . . . . . . 9 (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
149148, 2elrab2 3687 . . . . . . . 8 ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 ↔ ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ∧ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
150130, 146, 149sylanbrc 582 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾)
151130resexd 6029 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
1521, 105, 150, 151fvmptd3 7022 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
153 resundir 5997 . . . . . . 7 ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵))
154 fnresdm 6670 . . . . . . . . . 10 (𝑎 Fn 𝐵 → (𝑎𝐵) = 𝑎)
155108, 133, 1543syl 18 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎𝐵) = 𝑎)
156 ffn 6718 . . . . . . . . . . . . 13 ((𝐴 × { 0 }):𝐴⟶{ 0 } → (𝐴 × { 0 }) Fn 𝐴)
157 fnresdisj 6671 . . . . . . . . . . . . 13 ((𝐴 × { 0 }) Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅))
158110, 156, 157mp2b 10 . . . . . . . . . . . 12 ((𝐴𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
159158biimpi 215 . . . . . . . . . . 11 ((𝐴𝐵) = ∅ → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
1601593ad2ant3 1134 . . . . . . . . . 10 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
161160adantr 480 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
162155, 161uneq12d 4165 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = (𝑎 ∪ ∅))
163 un0 4391 . . . . . . . 8 (𝑎 ∪ ∅) = 𝑎
164162, 163eqtrdi 2787 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = 𝑎)
165153, 164eqtrid 2783 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = 𝑎)
166152, 165eqtrd 2771 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = 𝑎)
16795, 13lmhmf 20790 . . . . . . . 8 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:𝐾⟶(Base‘𝐷))
168 ffn 6718 . . . . . . . 8 (𝐹:𝐾⟶(Base‘𝐷) → 𝐹 Fn 𝐾)
16949, 167, 1683syl 18 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 Fn 𝐾)
170169adantr 480 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝐹 Fn 𝐾)
171 fnfvelrn 7083 . . . . . 6 ((𝐹 Fn 𝐾 ∧ (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
172170, 150, 171syl2anc 583 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
173166, 172eqeltrrd 2833 . . . 4 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎 ∈ ran 𝐹)
174104, 173eqelssd 4004 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ran 𝐹 = (Base‘𝐷))
175 dff1o5 6843 . . 3 (𝐹:𝐾1-1-onto→(Base‘𝐷) ↔ (𝐹:𝐾1-1→(Base‘𝐷) ∧ ran 𝐹 = (Base‘𝐷)))
176100, 174, 175sylanbrc 582 . 2 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹:𝐾1-1-onto→(Base‘𝐷))
17795, 13islmim 20818 . 2 (𝐹 ∈ (𝐿 LMIso 𝐷) ↔ (𝐹 ∈ (𝐿 LMHom 𝐷) ∧ 𝐹:𝐾1-1-onto→(Base‘𝐷)))
17849, 176, 177sylanbrc 582 1 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wral 3060  {crab 3431  Vcvv 3473  cun 3947  cin 3948  wss 3949  c0 4323  {csn 4629  cmpt 5232   × cxp 5675  ccnv 5676  ran crn 5678  cres 5679  cima 5680   Fn wfn 6539  wf 6540  1-1wf1 6541  1-1-ontowf1o 6543  cfv 6544  (class class class)co 7412  Basecbs 17149  s cress 17178  0gc0g 17390  s cpws 17397  Mndcmnd 18660  Grpcgrp 18856  SubGrpcsubg 19037   GrpHom cghm 19128  LModclmod 20615  LSubSpclss 20687   LMHom clmhm 20775   LMIso clmim 20776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-of 7673  df-om 7859  df-1st 7978  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-er 8706  df-map 8825  df-ixp 8895  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9440  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-nn 12218  df-2 12280  df-3 12281  df-4 12282  df-5 12283  df-6 12284  df-7 12285  df-8 12286  df-9 12287  df-n0 12478  df-z 12564  df-dec 12683  df-uz 12828  df-fz 13490  df-struct 17085  df-sets 17102  df-slot 17120  df-ndx 17132  df-base 17150  df-ress 17179  df-plusg 17215  df-mulr 17216  df-sca 17218  df-vsca 17219  df-ip 17220  df-tset 17221  df-ple 17222  df-ds 17224  df-hom 17226  df-cco 17227  df-0g 17392  df-prds 17398  df-pws 17400  df-mgm 18566  df-sgrp 18645  df-mnd 18661  df-grp 18859  df-minusg 18860  df-sbg 18861  df-subg 19040  df-ghm 19129  df-cmn 19692  df-abl 19693  df-mgp 20030  df-rng 20048  df-ur 20077  df-ring 20130  df-lmod 20617  df-lss 20688  df-lmhm 20778  df-lmim 20779
This theorem is referenced by:  pwslnmlem2  42138
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