Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pwssplit4 Structured version   Visualization version   GIF version

Theorem pwssplit4 36567
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 3554 . . . . . 6 {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })} ⊆ 𝐺
42, 3eqsstri 3502 . . . . 5 𝐾𝐺
5 resmpt 5260 . . . . 5 (𝐾𝐺 → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) = (𝑥𝐾 ↦ (𝑥𝐵)))
64, 5ax-mp 5 . . . 4 ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) = (𝑥𝐾 ↦ (𝑥𝐵))
71, 6eqtr4i 2539 . . 3 𝐹 = ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾)
8 ssun2 3643 . . . . . 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 2514 . . . . . 6 (Base‘𝐷) = (Base‘𝐷)
14 eqid 2514 . . . . . 6 (𝑥𝐺 ↦ (𝑥𝐵)) = (𝑥𝐺 ↦ (𝑥𝐵))
1510, 11, 12, 13, 14pwssplit3 18786 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉𝐵 ⊆ (𝐴𝐵)) → (𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷))
169, 15syld3an3 1362 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷))
17 simp1 1053 . . . . . . . . . 10 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝑅 ∈ LMod)
18 lmodgrp 18600 . . . . . . . . . 10 (𝑅 ∈ LMod → 𝑅 ∈ Grp)
19 grpmnd 17144 . . . . . . . . . 10 (𝑅 ∈ Grp → 𝑅 ∈ Mnd)
2017, 18, 193syl 18 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝑅 ∈ Mnd)
21 ssun1 3642 . . . . . . . . . . 11 𝐴 ⊆ (𝐴𝐵)
22 ssexg 4631 . . . . . . . . . . 11 ((𝐴 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐴 ∈ V)
2321, 22mpan 701 . . . . . . . . . 10 ((𝐴𝐵) ∈ 𝑉𝐴 ∈ V)
24233ad2ant2 1075 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐴 ∈ V)
25 pwssplit4.c . . . . . . . . . 10 𝐶 = (𝑅s 𝐴)
26 pwssplit4.z . . . . . . . . . 10 0 = (0g𝑅)
2725, 26pws0g 17041 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × { 0 }) = (0g𝐶))
2820, 24, 27syl2anc 690 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴 × { 0 }) = (0g𝐶))
2928eqeq2d 2524 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ (𝑦𝐴) = (0g𝐶)))
3029rabbidv 3068 . . . . . 6 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → {𝑦𝐺 ∣ (𝑦𝐴) = (𝐴 × { 0 })} = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
312, 30syl5eq 2560 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
3221a1i 11 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐴 ⊆ (𝐴𝐵))
33 eqid 2514 . . . . . . . 8 (Base‘𝐶) = (Base‘𝐶)
34 eqid 2514 . . . . . . . 8 (𝑦𝐺 ↦ (𝑦𝐴)) = (𝑦𝐺 ↦ (𝑦𝐴))
3510, 25, 12, 33, 34pwssplit3 18786 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉𝐴 ⊆ (𝐴𝐵)) → (𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶))
3632, 35syld3an3 1362 . . . . . 6 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑦𝐺 ↦ (𝑦𝐴)) ∈ (𝐸 LMHom 𝐶))
37 fvex 5997 . . . . . . . . 9 (0g𝐶) ∈ V
3834mptiniseg 5436 . . . . . . . . 9 ((0g𝐶) ∈ V → ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)}) = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)})
3937, 38ax-mp 5 . . . . . . . 8 ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)}) = {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)}
4039eqcomi 2523 . . . . . . 7 {𝑦𝐺 ∣ (𝑦𝐴) = (0g𝐶)} = ((𝑦𝐺 ↦ (𝑦𝐴)) “ {(0g𝐶)})
41 eqid 2514 . . . . . . 7 (0g𝐶) = (0g𝐶)
42 eqid 2514 . . . . . . 7 (LSubSp‘𝐸) = (LSubSp‘𝐸)
4340, 41, 42lmhmkerlss 18776 . . . . . 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 2592 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 ∈ (LSubSp‘𝐸))
46 pwssplit4.l . . . . 5 𝐿 = (𝐸s 𝐾)
4742, 46reslmhm 18777 . . . 4 (((𝑥𝐺 ↦ (𝑥𝐵)) ∈ (𝐸 LMHom 𝐷) ∧ 𝐾 ∈ (LSubSp‘𝐸)) → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
4816, 45, 47syl2anc 690 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑥𝐺 ↦ (𝑥𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
497, 48syl5eqel 2596 . 2 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMHom 𝐷))
501fvtresfn 6077 . . . . . . 7 (𝑎𝐾 → (𝐹𝑎) = (𝑎𝐵))
51 ssexg 4631 . . . . . . . . . . 11 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ∈ 𝑉) → 𝐵 ∈ V)
528, 51mpan 701 . . . . . . . . . 10 ((𝐴𝐵) ∈ 𝑉𝐵 ∈ V)
53523ad2ant2 1075 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐵 ∈ V)
5411, 26pws0g 17041 . . . . . . . . 9 ((𝑅 ∈ Mnd ∧ 𝐵 ∈ V) → (𝐵 × { 0 }) = (0g𝐷))
5520, 53, 54syl2anc 690 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐵 × { 0 }) = (0g𝐷))
5655eqcomd 2520 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (0g𝐷) = (𝐵 × { 0 }))
5750, 56eqeqan12rd 2532 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝐹𝑎) = (0g𝐷) ↔ (𝑎𝐵) = (𝐵 × { 0 })))
58 reseq1 5202 . . . . . . . . . 10 (𝑦 = 𝑎 → (𝑦𝐴) = (𝑎𝐴))
5958eqeq1d 2516 . . . . . . . . 9 (𝑦 = 𝑎 → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ (𝑎𝐴) = (𝐴 × { 0 })))
6059, 2elrab2 3237 . . . . . . . 8 (𝑎𝐾 ↔ (𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })))
61 uneq12 3628 . . . . . . . . . . . . 13 (((𝑎𝐴) = (𝐴 × { 0 }) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → ((𝑎𝐴) ∪ (𝑎𝐵)) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 })))
62 resundi 5221 . . . . . . . . . . . . 13 (𝑎 ↾ (𝐴𝐵)) = ((𝑎𝐴) ∪ (𝑎𝐵))
63 xpundir 4989 . . . . . . . . . . . . 13 ((𝐴𝐵) × { 0 }) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 }))
6461, 62, 633eqtr4g 2573 . . . . . . . . . . . 12 (((𝑎𝐴) = (𝐴 × { 0 }) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
6564adantll 745 . . . . . . . . . . 11 (((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
6665adantl 480 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → (𝑎 ↾ (𝐴𝐵)) = ((𝐴𝐵) × { 0 }))
67 eqid 2514 . . . . . . . . . . . 12 (Base‘𝑅) = (Base‘𝑅)
68 simpl1 1056 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑅 ∈ LMod)
69 simp2 1054 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) ∈ 𝑉)
7069adantr 479 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → (𝐴𝐵) ∈ 𝑉)
71 simprll 797 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎𝐺)
7210, 67, 12, 68, 70, 71pwselbas 15856 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎:(𝐴𝐵)⟶(Base‘𝑅))
73 ffn 5843 . . . . . . . . . . 11 (𝑎:(𝐴𝐵)⟶(Base‘𝑅) → 𝑎 Fn (𝐴𝐵))
74 fnresdm 5799 . . . . . . . . . . 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 17041 . . . . . . . . . . . . 13 ((𝑅 ∈ Mnd ∧ (𝐴𝐵) ∈ 𝑉) → ((𝐴𝐵) × { 0 }) = (0g𝐸))
7720, 69, 76syl2anc 690 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) × { 0 }) = (0g𝐸))
7810pwslmod 18695 . . . . . . . . . . . . . . 15 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉) → 𝐸 ∈ LMod)
79783adant3 1073 . . . . . . . . . . . . . 14 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐸 ∈ LMod)
8042lsssubg 18682 . . . . . . . . . . . . . 14 ((𝐸 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝐸)) → 𝐾 ∈ (SubGrp‘𝐸))
8179, 45, 80syl2anc 690 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐾 ∈ (SubGrp‘𝐸))
82 eqid 2514 . . . . . . . . . . . . . 14 (0g𝐸) = (0g𝐸)
8346, 82subg0 17315 . . . . . . . . . . . . 13 (𝐾 ∈ (SubGrp‘𝐸) → (0g𝐸) = (0g𝐿))
8481, 83syl 17 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (0g𝐸) = (0g𝐿))
8577, 84eqtrd 2548 . . . . . . . . . . 11 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) × { 0 }) = (0g𝐿))
8685adantr 479 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → ((𝐴𝐵) × { 0 }) = (0g𝐿))
8766, 75, 863eqtr3d 2556 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) ∧ (𝑎𝐵) = (𝐵 × { 0 }))) → 𝑎 = (0g𝐿))
8887exp32 628 . . . . . . . 8 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑎𝐺 ∧ (𝑎𝐴) = (𝐴 × { 0 })) → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿))))
8960, 88syl5bi 230 . . . . . . 7 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑎𝐾 → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿))))
9089imp 443 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝑎𝐵) = (𝐵 × { 0 }) → 𝑎 = (0g𝐿)))
9157, 90sylbid 228 . . . . 5 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎𝐾) → ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿)))
9291ralrimiva 2853 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ∀𝑎𝐾 ((𝐹𝑎) = (0g𝐷) → 𝑎 = (0g𝐿)))
93 lmghm 18756 . . . . 5 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹 ∈ (𝐿 GrpHom 𝐷))
9446, 12ressbas2 15642 . . . . . . 7 (𝐾𝐺𝐾 = (Base‘𝐿))
954, 94ax-mp 5 . . . . . 6 𝐾 = (Base‘𝐿)
96 eqid 2514 . . . . . 6 (0g𝐿) = (0g𝐿)
97 eqid 2514 . . . . . 6 (0g𝐷) = (0g𝐷)
9895, 13, 96, 97ghmf1 17404 . . . . 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 245 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹:𝐾1-1→(Base‘𝐷))
101 eqid 2514 . . . . . 6 (Base‘𝐿) = (Base‘𝐿)
102101, 13lmhmf 18759 . . . . 5 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:(Base‘𝐿)⟶(Base‘𝐷))
103 frn 5851 . . . . 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 15855 . . . . . . . . . . . . . . 15 ((𝑅 ∈ LMod ∧ 𝐵 ∈ V) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
10617, 53, 105syl2anc 690 . . . . . . . . . . . . . 14 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
107106biimpa 499 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎:𝐵⟶(Base‘𝑅))
108 fvex 5997 . . . . . . . . . . . . . . . . 17 (0g𝑅) ∈ V
10926, 108eqeltri 2588 . . . . . . . . . . . . . . . 16 0 ∈ V
110109fconst 5888 . . . . . . . . . . . . . . 15 (𝐴 × { 0 }):𝐴⟶{ 0 }
111110a1i 11 . . . . . . . . . . . . . 14 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶{ 0 })
11220adantr 479 . . . . . . . . . . . . . . . 16 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑅 ∈ Mnd)
11367, 26mndidcl 17023 . . . . . . . . . . . . . . . 16 (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅))
114112, 113syl 17 . . . . . . . . . . . . . . 15 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 0 ∈ (Base‘𝑅))
115114snssd 4184 . . . . . . . . . . . . . 14 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → { 0 } ⊆ (Base‘𝑅))
116111, 115fssd 5855 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶(Base‘𝑅))
117 incom 3670 . . . . . . . . . . . . . . 15 (𝐵𝐴) = (𝐴𝐵)
118 simp3 1055 . . . . . . . . . . . . . . 15 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐴𝐵) = ∅)
119117, 118syl5eq 2560 . . . . . . . . . . . . . 14 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝐵𝐴) = ∅)
120119adantr 479 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵𝐴) = ∅)
121 fun 5864 . . . . . . . . . . . . 13 (((𝑎:𝐵⟶(Base‘𝑅) ∧ (𝐴 × { 0 }):𝐴⟶(Base‘𝑅)) ∧ (𝐵𝐴) = ∅) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
122107, 116, 120, 121syl21anc 1316 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
123 uncom 3623 . . . . . . . . . . . . 13 (𝐵𝐴) = (𝐴𝐵)
124 unidm 3622 . . . . . . . . . . . . 13 ((Base‘𝑅) ∪ (Base‘𝑅)) = (Base‘𝑅)
125123, 124feq23i 5837 . . . . . . . . . . . 12 ((𝑎 ∪ (𝐴 × { 0 })):(𝐵𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)) ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅))
126122, 125sylib 206 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅))
12710, 67, 12pwselbasb 15855 . . . . . . . . . . . . 13 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
1281273adant3 1073 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
129128adantr 479 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴𝐵)⟶(Base‘𝑅)))
130126, 129mpbird 245 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺)
131 simpl3 1058 . . . . . . . . . . . . . 14 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴𝐵) = ∅)
132117, 131syl5eq 2560 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵𝐴) = ∅)
133 ffn 5843 . . . . . . . . . . . . . 14 (𝑎:𝐵⟶(Base‘𝑅) → 𝑎 Fn 𝐵)
134 fnresdisj 5800 . . . . . . . . . . . . . 14 (𝑎 Fn 𝐵 → ((𝐵𝐴) = ∅ ↔ (𝑎𝐴) = ∅))
135107, 133, 1343syl 18 . . . . . . . . . . . . 13 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐵𝐴) = ∅ ↔ (𝑎𝐴) = ∅))
136132, 135mpbid 220 . . . . . . . . . . . 12 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎𝐴) = ∅)
137 fnconstg 5890 . . . . . . . . . . . . . 14 ( 0 ∈ V → (𝐴 × { 0 }) Fn 𝐴)
138 fnresdm 5799 . . . . . . . . . . . . . 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 3634 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴)) = (∅ ∪ (𝐴 × { 0 })))
142 resundir 5222 . . . . . . . . . . 11 ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = ((𝑎𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴))
143 uncom 3623 . . . . . . . . . . . 12 (∅ ∪ (𝐴 × { 0 })) = ((𝐴 × { 0 }) ∪ ∅)
144 un0 3822 . . . . . . . . . . . 12 ((𝐴 × { 0 }) ∪ ∅) = (𝐴 × { 0 })
145143, 144eqtr2i 2537 . . . . . . . . . . 11 (𝐴 × { 0 }) = (∅ ∪ (𝐴 × { 0 }))
146141, 142, 1453eqtr4g 2573 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 }))
147 reseq1 5202 . . . . . . . . . . . 12 (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑦𝐴) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴))
148147eqeq1d 2516 . . . . . . . . . . 11 (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → ((𝑦𝐴) = (𝐴 × { 0 }) ↔ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
149148, 2elrab2 3237 . . . . . . . . . 10 ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 ↔ ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ∧ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
150130, 146, 149sylanbrc 694 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾)
151 resexg 5253 . . . . . . . . . 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 5202 . . . . . . . . . 10 (𝑥 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑥𝐵) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
154153, 1fvmptg 6073 . . . . . . . . 9 (((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 ∧ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
155150, 152, 154syl2anc 690 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
156 resundir 5222 . . . . . . . . 9 ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵))
157 fnresdm 5799 . . . . . . . . . . . 12 (𝑎 Fn 𝐵 → (𝑎𝐵) = 𝑎)
158107, 133, 1573syl 18 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎𝐵) = 𝑎)
159 ffn 5843 . . . . . . . . . . . . . . 15 ((𝐴 × { 0 }):𝐴⟶{ 0 } → (𝐴 × { 0 }) Fn 𝐴)
160 fnresdisj 5800 . . . . . . . . . . . . . . 15 ((𝐴 × { 0 }) Fn 𝐴 → ((𝐴𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅))
161110, 159, 160mp2b 10 . . . . . . . . . . . . . 14 ((𝐴𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
162161biimpi 204 . . . . . . . . . . . . 13 ((𝐴𝐵) = ∅ → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
1631623ad2ant3 1076 . . . . . . . . . . . 12 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
164163adantr 479 . . . . . . . . . . 11 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
165158, 164uneq12d 3634 . . . . . . . . . 10 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = (𝑎 ∪ ∅))
166 un0 3822 . . . . . . . . . 10 (𝑎 ∪ ∅) = 𝑎
167165, 166syl6eq 2564 . . . . . . . . 9 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = 𝑎)
168156, 167syl5eq 2560 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = 𝑎)
169155, 168eqtrd 2548 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = 𝑎)
17095, 13lmhmf 18759 . . . . . . . . . 10 (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:𝐾⟶(Base‘𝐷))
171 ffn 5843 . . . . . . . . . 10 (𝐹:𝐾⟶(Base‘𝐷) → 𝐹 Fn 𝐾)
17249, 170, 1713syl 18 . . . . . . . . 9 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 Fn 𝐾)
173172adantr 479 . . . . . . . 8 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝐹 Fn 𝐾)
174 fnfvelrn 6148 . . . . . . . 8 ((𝐹 Fn 𝐾 ∧ (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
175173, 150, 174syl2anc 690 . . . . . . 7 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
176169, 175eqeltrrd 2593 . . . . . 6 (((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎 ∈ ran 𝐹)
177176ex 448 . . . . 5 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (𝑎 ∈ (Base‘𝐷) → 𝑎 ∈ ran 𝐹))
178177ssrdv 3478 . . . 4 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → (Base‘𝐷) ⊆ ran 𝐹)
179104, 178eqssd 3489 . . 3 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → ran 𝐹 = (Base‘𝐷))
180 dff1o5 5943 . . 3 (𝐹:𝐾1-1-onto→(Base‘𝐷) ↔ (𝐹:𝐾1-1→(Base‘𝐷) ∧ ran 𝐹 = (Base‘𝐷)))
181100, 179, 180sylanbrc 694 . 2 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹:𝐾1-1-onto→(Base‘𝐷))
18295, 13islmim 18787 . 2 (𝐹 ∈ (𝐿 LMIso 𝐷) ↔ (𝐹 ∈ (𝐿 LMHom 𝐷) ∧ 𝐹:𝐾1-1-onto→(Base‘𝐷)))
18349, 181, 182sylanbrc 694 1 ((𝑅 ∈ LMod ∧ (𝐴𝐵) ∈ 𝑉 ∧ (𝐴𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1938  wral 2800  {crab 2804  Vcvv 3077  cun 3442  cin 3443  wss 3444  c0 3777  {csn 4028  cmpt 4541   × cxp 4930  ccnv 4931  ran crn 4933  cres 4934  cima 4935   Fn wfn 5684  wf 5685  1-1wf1 5686  1-1-ontowf1o 5688  cfv 5689  (class class class)co 6426  Basecbs 15579  s cress 15580  0gc0g 15807  s cpws 15814  Mndcmnd 17009  Grpcgrp 17137  SubGrpcsubg 17303   GrpHom cghm 17372  LModclmod 18593  LSubSpclss 18657   LMHom clmhm 18744   LMIso clmim 18745
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-cnex 9747  ax-resscn 9748  ax-1cn 9749  ax-icn 9750  ax-addcl 9751  ax-addrcl 9752  ax-mulcl 9753  ax-mulrcl 9754  ax-mulcom 9755  ax-addass 9756  ax-mulass 9757  ax-distr 9758  ax-i2m1 9759  ax-1ne0 9760  ax-1rid 9761  ax-rnegex 9762  ax-rrecex 9763  ax-cnre 9764  ax-pre-lttri 9765  ax-pre-lttrn 9766  ax-pre-ltadd 9767  ax-pre-mulgt0 9768
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-reu 2807  df-rmo 2808  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-int 4309  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-riota 6388  df-ov 6429  df-oprab 6430  df-mpt2 6431  df-of 6671  df-om 6834  df-1st 6934  df-2nd 6935  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-1o 7323  df-oadd 7327  df-er 7505  df-map 7622  df-ixp 7671  df-en 7718  df-dom 7719  df-sdom 7720  df-fin 7721  df-sup 8107  df-pnf 9831  df-mnf 9832  df-xr 9833  df-ltxr 9834  df-le 9835  df-sub 10019  df-neg 10020  df-nn 10776  df-2 10834  df-3 10835  df-4 10836  df-5 10837  df-6 10838  df-7 10839  df-8 10840  df-9 10841  df-n0 11048  df-z 11119  df-dec 11234  df-uz 11428  df-fz 12066  df-struct 15581  df-ndx 15582  df-slot 15583  df-base 15584  df-sets 15585  df-ress 15586  df-plusg 15665  df-mulr 15666  df-sca 15668  df-vsca 15669  df-ip 15670  df-tset 15671  df-ple 15672  df-ds 15675  df-hom 15677  df-cco 15678  df-0g 15809  df-prds 15815  df-pws 15817  df-mgm 16957  df-sgrp 16999  df-mnd 17010  df-grp 17140  df-minusg 17141  df-sbg 17142  df-subg 17306  df-ghm 17373  df-mgp 18220  df-ur 18232  df-ring 18279  df-lmod 18595  df-lss 18658  df-lmhm 18747  df-lmim 18748
This theorem is referenced by:  pwslnmlem2  36571
  Copyright terms: Public domain W3C validator