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Theorem pwssplit4 43101
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 4080 . . . . . 6 {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })} ⊆ 𝐺
42, 3eqsstri 4030 . . . . 5 𝐾𝐺
5 resmpt 6055 . . . . 5 (𝐾𝐺 → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) = (𝑥𝐾 ↦ (𝑥𝐵)))
64, 5ax-mp 5 . . . 4 ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) = (𝑥𝐾 ↦ (𝑥𝐵))
71, 6eqtr4i 2768 . . 3 𝐹 = ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾)
8 ssun2 4179 . . . . . 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 2737 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
14 eqid 2737 . . . . . 6 (𝑥𝐺 ↦ (𝑥𝐵)) = (𝑥𝐺 ↦ (𝑥𝐵))
1510, 11, 12, 13, 14pwssplit3 21060 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷))
169, 15syld3an3 1411 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷))
17 simp1 1137 . . . . . . . . . 10 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝑅 ∈ LMod)
18 lmodgrp 20865 . . . . . . . . . 10 (𝑅 ∈ LMod → 𝑅 ∈ Grp)
19 grpmnd 18958 . . . . . . . . . 10 (𝑅 ∈ Grp → 𝑅 ∈ Mnd)
2017, 18, 193syl 18 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝑅 ∈ Mnd)
21 ssun1 4178 . . . . . . . . . . 11 𝐴 ⊆ (𝐴𝐵)
22 ssexg 5323 . . . . . . . . . . 11 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐴 ∈ V)
2321, 22mpan 690 . . . . . . . . . 10 ((𝐴𝐵) ∈ 𝑉𝐴 ∈ V)
24233ad2ant2 1135 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐴 ∈ V)
25 pwssplit4.c . . . . . . . . . 10 𝐶 = (𝑅s 𝐴)
26 pwssplit4.z . . . . . . . . . 10 0 = (0g𝑅)
2725, 26pws0g 18786 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × { 0 }) = (0g𝐶))
2820, 24, 27syl2anc 584 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴 × { 0 }) = (0g𝐶))
2928eqeq2d 2748 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ (𝑦𝐴) = (0g𝐶)))
3029rabbidv 3444 . . . . . 6 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })} = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
312, 30eqtrid 2789 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
3221a1i 11 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ (𝐴𝐵))
33 eqid 2737 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
34 eqid 2737 . . . . . . . 8 (𝑦𝐺 ↦ (𝑦𝐴)) = (𝑦𝐺 ↦ (𝑦𝐴))
3510, 25, 12, 33, 34pwssplit3 21060 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉𝐴 ⊆ (𝐴𝐵)) → (𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶))
3632, 35syld3an3 1411 . . . . . 6 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶))
37 fvex 6919 . . . . . . . . 9 (0g𝐶) ∈ V
3834mptiniseg 6259 . . . . . . . . 9 ((0g𝐶) ∈ V → ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)}) = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
3937, 38ax-mp 5 . . . . . . . 8 ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)}) = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)}
4039eqcomi 2746 . . . . . . 7 {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)} = ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)})
41 eqid 2737 . . . . . . 7 (0g𝐶) = (0g𝐶)
42 eqid 2737 . . . . . . 7 (LSubSp‘𝐸) = (LSubSp‘𝐸)
4340, 41, 42lmhmkerlss 21050 . . . . . 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 2841 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 ∈ (LSubSp‘𝐸))
46 pwssplit4.l . . . . 5 𝐿 = (𝐸s 𝐾)
4742, 46reslmhm 21051 . . . 4 (((𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷) ∧ 𝐾 ∈ (LSubSp‘𝐸)) → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
4816, 45, 47syl2anc 584 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
497, 48eqeltrid 2845 . 2 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMHom 𝐷))
501fvtresfn 7018 . . . . . . 7 (𝑎𝐾 → (𝐹𝑎) = (𝑎𝐵))
51 ssexg 5323 . . . . . . . . . . 11 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐵 ∈ V)
528, 51mpan 690 . . . . . . . . . 10 ((𝐴𝐵) ∈ 𝑉𝐵 ∈ V)
53523ad2ant2 1135 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐵 ∈ V)
5411, 26pws0g 18786 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐵 ∈ V) → (𝐵 × { 0 }) = (0g𝐷))
5520, 53, 54syl2anc 584 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐵 × { 0 }) = (0g𝐷))
5655eqcomd 2743 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (0g𝐷) = (𝐵 × { 0 }))
5750, 56eqeqan12rd 2752 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝐹𝑎) = (0g𝐷) ↔ (𝑎𝐵) = (𝐵 × { 0 })))
58 reseq1 5991 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝑦𝐴) = (𝑎𝐴))
5958eqeq1d 2739 . . . . . . . . 9 (𝑦 = 𝑎 → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ (𝑎𝐴) = (𝐴 × { 0 })))
6059, 2elrab2 3695 . . . . . . . 8 (𝑎𝐾 ↔ (𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })))
61 uneq12 4163 . . . . . . . . . . . . 13 (((𝑎𝐴) = (𝐴 × { 0 }) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → ((𝑎𝐴) ∪ (𝑎𝐵)) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 })))
62 resundi 6011 . . . . . . . . . . . . 13 (𝑎 ↾ (𝐴𝐵)) = ((𝑎𝐴) ∪ (𝑎𝐵))
63 xpundir 5755 . . . . . . . . . . . . 13 ((𝐴𝐵) × { 0 }) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 }))
6461, 62, 633eqtr4g 2802 . . . . . . . . . . . 12 (((𝑎𝐴) = (𝐴 × { 0 }) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
6564adantll 714 . . . . . . . . . . 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 2737 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
68 simpl1 1192 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑅 ∈ LMod)
69 simp2 1138 . . . . . . . . . . . . 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 779 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎𝐺)
7210, 67, 12, 68, 70, 71pwselbas 17534 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎:(𝐴𝐵)⟶(Base‘𝑅))
73 ffn 6736 . . . . . . . . . . 11 (𝑎:(𝐴𝐵)⟶(Base‘𝑅) → 𝑎 Fn (𝐴𝐵))
74 fnresdm 6687 . . . . . . . . . . 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 18786 . . . . . . . . . . . . 13 ((𝑅 ∈ Mnd ∧ (𝐴𝐵) ∈ 𝑉) → ((𝐴𝐵) × { 0 }) = (0g𝐸))
7720, 69, 76syl2anc 584 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) × { 0 }) = (0g𝐸))
7810pwslmod 20968 . . . . . . . . . . . . . . 15 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉) → 𝐸 ∈ LMod)
79783adant3 1133 . . . . . . . . . . . . . 14 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐸 ∈ LMod)
8042lsssubg 20955 . . . . . . . . . . . . . 14 ((𝐸 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝐸)) → 𝐾 ∈ (SubGrp‘𝐸))
8179, 45, 80syl2anc 584 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 ∈ (SubGrp‘𝐸))
82 eqid 2737 . . . . . . . . . . . . . 14 (0g𝐸) = (0g𝐸)
8346, 82subg0 19150 . . . . . . . . . . . . 13 (𝐾 ∈ (SubGrp‘𝐸) → (0g𝐸) = (0g𝐿))
8481, 83syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (0g𝐸) = (0g𝐿))
8577, 84eqtrd 2777 . . . . . . . . . . 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 2785 . . . . . . . . 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 242 . . . . . . 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 240 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿)))
9291ralrimiva 3146 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ∀𝑎𝐾 ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿)))
93 lmghm 21030 . . . . 5 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹 ∈ (𝐿 GrpHom 𝐷))
9446, 12ressbas2 17283 . . . . . . 7 (𝐾𝐺𝐾 = (Base‘𝐿))
954, 94ax-mp 5 . . . . . 6 𝐾 = (Base‘𝐿)
96 eqid 2737 . . . . . 6 (0g𝐿) = (0g𝐿)
97 eqid 2737 . . . . . 6 (0g𝐷) = (0g𝐷)
9895, 13, 96, 97ghmf1 19264 . . . . 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 257 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹:𝐾1-1→(Base‘𝐷))
101 eqid 2737 . . . . . 6 (Base‘𝐿) = (Base‘𝐿)
102101, 13lmhmf 21033 . . . . 5 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:(Base‘𝐿)⟶(Base‘𝐷))
103 frn 6743 . . . . 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 5991 . . . . . . 7 (𝑥 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑥𝐵) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
10611, 67, 13pwselbasb 17533 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ 𝐵 ∈ V) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
10717, 53, 106syl2anc 584 . . . . . . . . . . . 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 6920 . . . . . . . . . . . . . 14 0 ∈ V
110109fconst 6794 . . . . . . . . . . . . 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 18762 . . . . . . . . . . . . . 14 (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅))
114112, 113syl 17 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 0 ∈ (Base‘𝑅))
115114snssd 4809 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → { 0 } ⊆ (Base‘𝑅))
116111, 115fssd 6753 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶(Base‘𝑅))
117 incom 4209 . . . . . . . . . . . . 13 (𝐵𝐴) = (𝐴𝐵)
118 simp3 1139 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
119117, 118eqtrid 2789 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐵𝐴) = ∅)
120119adantr 480 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵𝐴) = ∅)
121 fun 6770 . . . . . . . . . . 11 (((𝑎:𝐵⟶(Base‘𝑅) ∧ (𝐴 × { 0 }):𝐴⟶(Base‘𝑅)) ∧ (𝐵𝐴) = ∅) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
122108, 116, 120, 121syl21anc 838 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
123 uncom 4158 . . . . . . . . . . 11 (𝐵𝐴) = (𝐴𝐵)
124 unidm 4157 . . . . . . . . . . 11 ((Base‘𝑅) ∪ (Base‘𝑅)) = (Base‘𝑅)
125123, 124feq23i 6730 . . . . . . . . . 10 ((𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)) ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅))
126122, 125sylib 218 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅))
12710, 67, 12pwselbasb 17533 . . . . . . . . . . 11 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
1281273adant3 1133 . . . . . . . . . 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 257 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺)
131 simpl3 1194 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴𝐵) = ∅)
132117, 131eqtrid 2789 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵𝐴) = ∅)
133 ffn 6736 . . . . . . . . . . . 12 (𝑎:𝐵⟶(Base‘𝑅) → 𝑎 Fn 𝐵)
134 fnresdisj 6688 . . . . . . . . . . . 12 (𝑎 Fn 𝐵 → ((𝐵𝐴) = ∅ ↔ (𝑎𝐴) = ∅))
135108, 133, 1343syl 18 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐵𝐴) = ∅ ↔ (𝑎𝐴) = ∅))
136132, 135mpbid 232 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎𝐴) = ∅)
137 fnconstg 6796 . . . . . . . . . . . 12 ( 0 ∈ V → (𝐴 × { 0 }) Fn 𝐴)
138 fnresdm 6687 . . . . . . . . . . . 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 4169 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴)) = (∅ ∪ (𝐴 × { 0 })))
142 resundir 6012 . . . . . . . . 9 ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = ((𝑎𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴))
143 uncom 4158 . . . . . . . . . 10 (∅ ∪ (𝐴 × { 0 })) = ((𝐴 × { 0 }) ∪ ∅)
144 un0 4394 . . . . . . . . . 10 ((𝐴 × { 0 }) ∪ ∅) = (𝐴 × { 0 })
145143, 144eqtr2i 2766 . . . . . . . . 9 (𝐴 × { 0 }) = (∅ ∪ (𝐴 × { 0 }))
146141, 142, 1453eqtr4g 2802 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 }))
147 reseq1 5991 . . . . . . . . . 10 (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑦𝐴) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴))
148147eqeq1d 2739 . . . . . . . . 9 (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
149148, 2elrab2 3695 . . . . . . . 8 ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 ↔ ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ∧ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
150130, 146, 149sylanbrc 583 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾)
151130resexd 6046 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
1521, 105, 150, 151fvmptd3 7039 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
153 resundir 6012 . . . . . . 7 ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵))
154 fnresdm 6687 . . . . . . . . . 10 (𝑎 Fn 𝐵 → (𝑎𝐵) = 𝑎)
155108, 133, 1543syl 18 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎𝐵) = 𝑎)
156 ffn 6736 . . . . . . . . . . . . 13 ((𝐴 × { 0 }):𝐴⟶{ 0 } → (𝐴 × { 0 }) Fn 𝐴)
157 fnresdisj 6688 . . . . . . . . . . . . 13 ((𝐴 × { 0 }) Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅))
158110, 156, 157mp2b 10 . . . . . . . . . . . 12 ((𝐴𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
159158biimpi 216 . . . . . . . . . . 11 ((𝐴𝐵) = ∅ → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
1601593ad2ant3 1136 . . . . . . . . . 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 4169 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = (𝑎 ∪ ∅))
163 un0 4394 . . . . . . . 8 (𝑎 ∪ ∅) = 𝑎
164162, 163eqtrdi 2793 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = 𝑎)
165153, 164eqtrid 2789 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = 𝑎)
166152, 165eqtrd 2777 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = 𝑎)
16795, 13lmhmf 21033 . . . . . . . 8 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:𝐾⟶(Base‘𝐷))
168 ffn 6736 . . . . . . . 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 7100 . . . . . 6 ((𝐹 Fn 𝐾 ∧ (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
172170, 150, 171syl2anc 584 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
173166, 172eqeltrrd 2842 . . . 4 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎 ∈ ran 𝐹)
174104, 173eqelssd 4005 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ran 𝐹 = (Base‘𝐷))
175 dff1o5 6857 . . 3 (𝐹:𝐾1-1-onto→(Base‘𝐷) ↔ (𝐹:𝐾1-1→(Base‘𝐷) ∧ ran 𝐹 = (Base‘𝐷)))
176100, 174, 175sylanbrc 583 . 2 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹:𝐾1-1-onto→(Base‘𝐷))
17795, 13islmim 21061 . 2 (𝐹 ∈ (𝐿 LMIso 𝐷) ↔ (𝐹 ∈ (𝐿 LMHom 𝐷) ∧ 𝐹:𝐾1-1-onto→(Base‘𝐷)))
17849, 176, 177sylanbrc 583 1 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436  Vcvv 3480  cun 3949  cin 3950  wss 3951  c0 4333  {csn 4626  cmpt 5225   × cxp 5683  ccnv 5684  ran crn 5686  cres 5687  cima 5688   Fn wfn 6556  wf 6557  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  0gc0g 17484  s cpws 17491  Mndcmnd 18747  Grpcgrp 18951  SubGrpcsubg 19138   GrpHom cghm 19230  LModclmod 20858  LSubSpclss 20929   LMHom clmhm 21018   LMIso clmim 21019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-fz 13548  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-0g 17486  df-prds 17492  df-pws 17494  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956  df-subg 19141  df-ghm 19231  df-cmn 19800  df-abl 19801  df-mgp 20138  df-rng 20150  df-ur 20179  df-ring 20232  df-lmod 20860  df-lss 20930  df-lmhm 21021  df-lmim 21022
This theorem is referenced by:  pwslnmlem2  43105
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