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Theorem pwssplit4 39865
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 4040 . . . . . 6 {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })} ⊆ 𝐺
42, 3eqsstri 3985 . . . . 5 𝐾𝐺
5 resmpt 5886 . . . . 5 (𝐾𝐺 → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) = (𝑥𝐾 ↦ (𝑥𝐵)))
64, 5ax-mp 5 . . . 4 ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) = (𝑥𝐾 ↦ (𝑥𝐵))
71, 6eqtr4i 2850 . . 3 𝐹 = ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾)
8 ssun2 4133 . . . . . 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 2824 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
14 eqid 2824 . . . . . 6 (𝑥𝐺 ↦ (𝑥𝐵)) = (𝑥𝐺 ↦ (𝑥𝐵))
1510, 11, 12, 13, 14pwssplit3 19819 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷))
169, 15syld3an3 1406 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷))
17 simp1 1133 . . . . . . . . . 10 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝑅 ∈ LMod)
18 lmodgrp 19627 . . . . . . . . . 10 (𝑅 ∈ LMod → 𝑅 ∈ Grp)
19 grpmnd 18099 . . . . . . . . . 10 (𝑅 ∈ Grp → 𝑅 ∈ Mnd)
2017, 18, 193syl 18 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝑅 ∈ Mnd)
21 ssun1 4132 . . . . . . . . . . 11 𝐴 ⊆ (𝐴𝐵)
22 ssexg 5208 . . . . . . . . . . 11 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐴 ∈ V)
2321, 22mpan 689 . . . . . . . . . 10 ((𝐴𝐵) ∈ 𝑉𝐴 ∈ V)
24233ad2ant2 1131 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐴 ∈ V)
25 pwssplit4.c . . . . . . . . . 10 𝐶 = (𝑅s 𝐴)
26 pwssplit4.z . . . . . . . . . 10 0 = (0g𝑅)
2725, 26pws0g 17936 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × { 0 }) = (0g𝐶))
2820, 24, 27syl2anc 587 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴 × { 0 }) = (0g𝐶))
2928eqeq2d 2835 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ (𝑦𝐴) = (0g𝐶)))
3029rabbidv 3465 . . . . . 6 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })} = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
312, 30syl5eq 2871 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
3221a1i 11 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ (𝐴𝐵))
33 eqid 2824 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
34 eqid 2824 . . . . . . . 8 (𝑦𝐺 ↦ (𝑦𝐴)) = (𝑦𝐺 ↦ (𝑦𝐴))
3510, 25, 12, 33, 34pwssplit3 19819 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉𝐴 ⊆ (𝐴𝐵)) → (𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶))
3632, 35syld3an3 1406 . . . . . 6 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶))
37 fvex 6664 . . . . . . . . 9 (0g𝐶) ∈ V
3834mptiniseg 6074 . . . . . . . . 9 ((0g𝐶) ∈ V → ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)}) = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
3937, 38ax-mp 5 . . . . . . . 8 ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)}) = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)}
4039eqcomi 2833 . . . . . . 7 {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)} = ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)})
41 eqid 2824 . . . . . . 7 (0g𝐶) = (0g𝐶)
42 eqid 2824 . . . . . . 7 (LSubSp‘𝐸) = (LSubSp‘𝐸)
4340, 41, 42lmhmkerlss 19809 . . . . . 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 2916 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 ∈ (LSubSp‘𝐸))
46 pwssplit4.l . . . . 5 𝐿 = (𝐸s 𝐾)
4742, 46reslmhm 19810 . . . 4 (((𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷) ∧ 𝐾 ∈ (LSubSp‘𝐸)) → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
4816, 45, 47syl2anc 587 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
497, 48eqeltrid 2920 . 2 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMHom 𝐷))
501fvtresfn 6751 . . . . . . 7 (𝑎𝐾 → (𝐹𝑎) = (𝑎𝐵))
51 ssexg 5208 . . . . . . . . . . 11 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐵 ∈ V)
528, 51mpan 689 . . . . . . . . . 10 ((𝐴𝐵) ∈ 𝑉𝐵 ∈ V)
53523ad2ant2 1131 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐵 ∈ V)
5411, 26pws0g 17936 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐵 ∈ V) → (𝐵 × { 0 }) = (0g𝐷))
5520, 53, 54syl2anc 587 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐵 × { 0 }) = (0g𝐷))
5655eqcomd 2830 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (0g𝐷) = (𝐵 × { 0 }))
5750, 56eqeqan12rd 2843 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝐹𝑎) = (0g𝐷) ↔ (𝑎𝐵) = (𝐵 × { 0 })))
58 reseq1 5828 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝑦𝐴) = (𝑎𝐴))
5958eqeq1d 2826 . . . . . . . . 9 (𝑦 = 𝑎 → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ (𝑎𝐴) = (𝐴 × { 0 })))
6059, 2elrab2 3668 . . . . . . . 8 (𝑎𝐾 ↔ (𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })))
61 uneq12 4118 . . . . . . . . . . . . 13 (((𝑎𝐴) = (𝐴 × { 0 }) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → ((𝑎𝐴) ∪ (𝑎𝐵)) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 })))
62 resundi 5848 . . . . . . . . . . . . 13 (𝑎 ↾ (𝐴𝐵)) = ((𝑎𝐴) ∪ (𝑎𝐵))
63 xpundir 5602 . . . . . . . . . . . . 13 ((𝐴𝐵) × { 0 }) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 }))
6461, 62, 633eqtr4g 2884 . . . . . . . . . . . 12 (((𝑎𝐴) = (𝐴 × { 0 }) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
6564adantll 713 . . . . . . . . . . 11 (((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
6665adantl 485 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
67 eqid 2824 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
68 simpl1 1188 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑅 ∈ LMod)
69 simp2 1134 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ 𝑉)
7069adantr 484 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → (𝐴𝐵) ∈ 𝑉)
71 simprll 778 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎𝐺)
7210, 67, 12, 68, 70, 71pwselbas 16751 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎:(𝐴𝐵)⟶(Base‘𝑅))
73 ffn 6495 . . . . . . . . . . 11 (𝑎:(𝐴𝐵)⟶(Base‘𝑅) → 𝑎 Fn (𝐴𝐵))
74 fnresdm 6447 . . . . . . . . . . 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 17936 . . . . . . . . . . . . 13 ((𝑅 ∈ Mnd ∧ (𝐴𝐵) ∈ 𝑉) → ((𝐴𝐵) × { 0 }) = (0g𝐸))
7720, 69, 76syl2anc 587 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) × { 0 }) = (0g𝐸))
7810pwslmod 19728 . . . . . . . . . . . . . . 15 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉) → 𝐸 ∈ LMod)
79783adant3 1129 . . . . . . . . . . . . . 14 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐸 ∈ LMod)
8042lsssubg 19715 . . . . . . . . . . . . . 14 ((𝐸 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝐸)) → 𝐾 ∈ (SubGrp‘𝐸))
8179, 45, 80syl2anc 587 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 ∈ (SubGrp‘𝐸))
82 eqid 2824 . . . . . . . . . . . . . 14 (0g𝐸) = (0g𝐸)
8346, 82subg0 18274 . . . . . . . . . . . . 13 (𝐾 ∈ (SubGrp‘𝐸) → (0g𝐸) = (0g𝐿))
8481, 83syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (0g𝐸) = (0g𝐿))
8577, 84eqtrd 2859 . . . . . . . . . . 11 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) × { 0 }) = (0g𝐿))
8685adantr 484 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → ((𝐴𝐵) × { 0 }) = (0g𝐿))
8766, 75, 863eqtr3d 2867 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎 = (0g𝐿))
8887exp32 424 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿))))
8960, 88syl5bi 245 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑎𝐾 → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿))))
9089imp 410 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿)))
9157, 90sylbid 243 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿)))
9291ralrimiva 3176 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ∀𝑎𝐾 ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿)))
93 lmghm 19789 . . . . 5 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹 ∈ (𝐿 GrpHom 𝐷))
9446, 12ressbas2 16544 . . . . . . 7 (𝐾𝐺𝐾 = (Base‘𝐿))
954, 94ax-mp 5 . . . . . 6 𝐾 = (Base‘𝐿)
96 eqid 2824 . . . . . 6 (0g𝐿) = (0g𝐿)
97 eqid 2824 . . . . . 6 (0g𝐷) = (0g𝐷)
9895, 13, 96, 97ghmf1 18376 . . . . 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 260 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹:𝐾1-1→(Base‘𝐷))
101 eqid 2824 . . . . . 6 (Base‘𝐿) = (Base‘𝐿)
102101, 13lmhmf 19792 . . . . 5 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:(Base‘𝐿)⟶(Base‘𝐷))
103 frn 6501 . . . . 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 5828 . . . . . . 7 (𝑥 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑥𝐵) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
10611, 67, 13pwselbasb 16750 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ 𝐵 ∈ V) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
10717, 53, 106syl2anc 587 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
108107biimpa 480 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎:𝐵⟶(Base‘𝑅))
10926fvexi 6665 . . . . . . . . . . . . . 14 0 ∈ V
110109fconst 6546 . . . . . . . . . . . . 13 (𝐴 × { 0 }):𝐴⟶{ 0 }
111110a1i 11 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶{ 0 })
11220adantr 484 . . . . . . . . . . . . . 14 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑅 ∈ Mnd)
11367, 26mndidcl 17915 . . . . . . . . . . . . . 14 (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅))
114112, 113syl 17 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 0 ∈ (Base‘𝑅))
115114snssd 4723 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → { 0 } ⊆ (Base‘𝑅))
116111, 115fssd 6509 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶(Base‘𝑅))
117 incom 4161 . . . . . . . . . . . . 13 (𝐵𝐴) = (𝐴𝐵)
118 simp3 1135 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
119117, 118syl5eq 2871 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐵𝐴) = ∅)
120119adantr 484 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵𝐴) = ∅)
121 fun 6521 . . . . . . . . . . 11 (((𝑎:𝐵⟶(Base‘𝑅) ∧ (𝐴 × { 0 }):𝐴⟶(Base‘𝑅)) ∧ (𝐵𝐴) = ∅) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
122108, 116, 120, 121syl21anc 836 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
123 uncom 4113 . . . . . . . . . . 11 (𝐵𝐴) = (𝐴𝐵)
124 unidm 4112 . . . . . . . . . . 11 ((Base‘𝑅) ∪ (Base‘𝑅)) = (Base‘𝑅)
125123, 124feq23i 6489 . . . . . . . . . 10 ((𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)) ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅))
126122, 125sylib 221 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅))
12710, 67, 12pwselbasb 16750 . . . . . . . . . . 11 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
1281273adant3 1129 . . . . . . . . . 10 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
129128adantr 484 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
130126, 129mpbird 260 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺)
131 simpl3 1190 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴𝐵) = ∅)
132117, 131syl5eq 2871 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵𝐴) = ∅)
133 ffn 6495 . . . . . . . . . . . 12 (𝑎:𝐵⟶(Base‘𝑅) → 𝑎 Fn 𝐵)
134 fnresdisj 6448 . . . . . . . . . . . 12 (𝑎 Fn 𝐵 → ((𝐵𝐴) = ∅ ↔ (𝑎𝐴) = ∅))
135108, 133, 1343syl 18 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐵𝐴) = ∅ ↔ (𝑎𝐴) = ∅))
136132, 135mpbid 235 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎𝐴) = ∅)
137 fnconstg 6548 . . . . . . . . . . . 12 ( 0 ∈ V → (𝐴 × { 0 }) Fn 𝐴)
138 fnresdm 6447 . . . . . . . . . . . 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 4124 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴)) = (∅ ∪ (𝐴 × { 0 })))
142 resundir 5849 . . . . . . . . 9 ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = ((𝑎𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴))
143 uncom 4113 . . . . . . . . . 10 (∅ ∪ (𝐴 × { 0 })) = ((𝐴 × { 0 }) ∪ ∅)
144 un0 4325 . . . . . . . . . 10 ((𝐴 × { 0 }) ∪ ∅) = (𝐴 × { 0 })
145143, 144eqtr2i 2848 . . . . . . . . 9 (𝐴 × { 0 }) = (∅ ∪ (𝐴 × { 0 }))
146141, 142, 1453eqtr4g 2884 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 }))
147 reseq1 5828 . . . . . . . . . 10 (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑦𝐴) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴))
148147eqeq1d 2826 . . . . . . . . 9 (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
149148, 2elrab2 3668 . . . . . . . 8 ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 ↔ ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ∧ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
150130, 146, 149sylanbrc 586 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾)
151 resexg 5879 . . . . . . . 8 ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
152150, 151syl 17 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
1531, 105, 150, 152fvmptd3 6772 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
154 resundir 5849 . . . . . . 7 ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵))
155 fnresdm 6447 . . . . . . . . . 10 (𝑎 Fn 𝐵 → (𝑎𝐵) = 𝑎)
156108, 133, 1553syl 18 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎𝐵) = 𝑎)
157 ffn 6495 . . . . . . . . . . . . 13 ((𝐴 × { 0 }):𝐴⟶{ 0 } → (𝐴 × { 0 }) Fn 𝐴)
158 fnresdisj 6448 . . . . . . . . . . . . 13 ((𝐴 × { 0 }) Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅))
159110, 157, 158mp2b 10 . . . . . . . . . . . 12 ((𝐴𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
160159biimpi 219 . . . . . . . . . . 11 ((𝐴𝐵) = ∅ → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
1611603ad2ant3 1132 . . . . . . . . . 10 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
162161adantr 484 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
163156, 162uneq12d 4124 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = (𝑎 ∪ ∅))
164 un0 4325 . . . . . . . 8 (𝑎 ∪ ∅) = 𝑎
165163, 164syl6eq 2875 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = 𝑎)
166154, 165syl5eq 2871 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = 𝑎)
167153, 166eqtrd 2859 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = 𝑎)
16895, 13lmhmf 19792 . . . . . . . 8 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:𝐾⟶(Base‘𝐷))
169 ffn 6495 . . . . . . . 8 (𝐹:𝐾⟶(Base‘𝐷) → 𝐹 Fn 𝐾)
17049, 168, 1693syl 18 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 Fn 𝐾)
171170adantr 484 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝐹 Fn 𝐾)
172 fnfvelrn 6829 . . . . . 6 ((𝐹 Fn 𝐾 ∧ (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
173171, 150, 172syl2anc 587 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
174167, 173eqeltrrd 2917 . . . 4 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎 ∈ ran 𝐹)
175104, 174eqelssd 3972 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ran 𝐹 = (Base‘𝐷))
176 dff1o5 6605 . . 3 (𝐹:𝐾1-1-onto→(Base‘𝐷) ↔ (𝐹:𝐾1-1→(Base‘𝐷) ∧ ran 𝐹 = (Base‘𝐷)))
177100, 175, 176sylanbrc 586 . 2 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹:𝐾1-1-onto→(Base‘𝐷))
17895, 13islmim 19820 . 2 (𝐹 ∈ (𝐿 LMIso 𝐷) ↔ (𝐹 ∈ (𝐿 LMHom 𝐷) ∧ 𝐹:𝐾1-1-onto→(Base‘𝐷)))
17949, 177, 178sylanbrc 586 1 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3132  {crab 3136  Vcvv 3479  cun 3916  cin 3917  wss 3918  c0 4274  {csn 4548  cmpt 5127   × cxp 5534  ccnv 5535  ran crn 5537  cres 5538  cima 5539   Fn wfn 6331  wf 6332  1-1wf1 6333  1-1-ontowf1o 6335  cfv 6336  (class class class)co 7138  Basecbs 16472  s cress 16473  0gc0g 16702  s cpws 16709  Mndcmnd 17900  Grpcgrp 18092  SubGrpcsubg 18262   GrpHom cghm 18344  LModclmod 19620  LSubSpclss 19689   LMHom clmhm 19777   LMIso clmim 19778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5171  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rmo 3140  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-of 7392  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-ixp 8445  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-sup 8890  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-nn 11624  df-2 11686  df-3 11687  df-4 11688  df-5 11689  df-6 11690  df-7 11691  df-8 11692  df-9 11693  df-n0 11884  df-z 11968  df-dec 12085  df-uz 12230  df-fz 12884  df-struct 16474  df-ndx 16475  df-slot 16476  df-base 16478  df-sets 16479  df-ress 16480  df-plusg 16567  df-mulr 16568  df-sca 16570  df-vsca 16571  df-ip 16572  df-tset 16573  df-ple 16574  df-ds 16576  df-hom 16578  df-cco 16579  df-0g 16704  df-prds 16710  df-pws 16712  df-mgm 17841  df-sgrp 17890  df-mnd 17901  df-grp 18095  df-minusg 18096  df-sbg 18097  df-subg 18265  df-ghm 18345  df-mgp 19229  df-ur 19241  df-ring 19288  df-lmod 19622  df-lss 19690  df-lmhm 19780  df-lmim 19781
This theorem is referenced by:  pwslnmlem2  39869
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