pwssplit4 - Mathbox for Stefan O'Rear

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Theorem pwssplit4 43331

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 = (𝑅 ↑_s (𝐴 ∪ 𝐵))
pwssplit4.g	⊢ 𝐺 = (Base‘𝐸)
pwssplit4.z	⊢ 0 = (0_g‘𝑅)
pwssplit4.k	⊢ 𝐾 = {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })}
pwssplit4.f	⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ↾ 𝐵))
pwssplit4.c	⊢ 𝐶 = (𝑅 ↑_s 𝐴)
pwssplit4.d	⊢ 𝐷 = (𝑅 ↑_s 𝐵)
pwssplit4.l	⊢ 𝐿 = (𝐸 ↾_s 𝐾)

**Assertion**
Ref	Expression
pwssplit4	⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 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.

Step	Hyp	Ref	Expression
1		pwssplit4.f	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐾 ↦ (𝑥 ↾ 𝐵))
2		pwssplit4.k	. . . . . 6 ⊢ 𝐾 = {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })}
3		ssrab2 4032	. . . . . 6 ⊢ {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })} ⊆ 𝐺
4	2, 3	eqsstri 3980	. . . . 5 ⊢ 𝐾 ⊆ 𝐺
5		resmpt 5996	. . . . 5 ⊢ (𝐾 ⊆ 𝐺 → ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾) = (𝑥 ∈ 𝐾 ↦ (𝑥 ↾ 𝐵)))
6	4, 5	ax-mp 5	. . . 4 ⊢ ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾) = (𝑥 ∈ 𝐾 ↦ (𝑥 ↾ 𝐵))
7	1, 6	eqtr4i 2762	. . 3 ⊢ 𝐹 = ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾)
8		ssun2 4131	. . . . . 6 ⊢ 𝐵 ⊆ (𝐴 ∪ 𝐵)
9	8	a1i 11	. . . . 5 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ⊆ (𝐴 ∪ 𝐵))
10		pwssplit4.e	. . . . . 6 ⊢ 𝐸 = (𝑅 ↑_s (𝐴 ∪ 𝐵))
11		pwssplit4.d	. . . . . 6 ⊢ 𝐷 = (𝑅 ↑_s 𝐵)
12		pwssplit4.g	. . . . . 6 ⊢ 𝐺 = (Base‘𝐸)
13		eqid 2736	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
14		eqid 2736	. . . . . 6 ⊢ (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) = (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵))
15	10, 11, 12, 13, 14	pwssplit3 21013	. . . . 5 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐸 LMHom 𝐷))
16	9, 15	syld3an3 1411	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐸 LMHom 𝐷))
17		simp1 1136	. . . . . . . . . 10 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 ∈ LMod)
18		lmodgrp 20818	. . . . . . . . . 10 ⊢ (𝑅 ∈ LMod → 𝑅 ∈ Grp)
19		grpmnd 18870	. . . . . . . . . 10 ⊢ (𝑅 ∈ Grp → 𝑅 ∈ Mnd)
20	17, 18, 19	3syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 ∈ Mnd)
21		ssun1 4130	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
22		ssexg 5268	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
23	21, 22	mpan 690	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝑉 → 𝐴 ∈ V)
24	23	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ∈ V)
25		pwssplit4.c	. . . . . . . . . 10 ⊢ 𝐶 = (𝑅 ↑_s 𝐴)
26		pwssplit4.z	. . . . . . . . . 10 ⊢ 0 = (0_g‘𝑅)
27	25, 26	pws0g 18698	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × { 0 }) = (0_g‘𝐶))
28	20, 24, 27	syl2anc 584	. . . . . . . 8 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 × { 0 }) = (0_g‘𝐶))
29	28	eqeq2d 2747	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ (𝑦 ↾ 𝐴) = (0_g‘𝐶)))
30	29	rabbidv 3406	. . . . . 6 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })} = {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)})
31	2, 30	eqtrid 2783	. . . . 5 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐾 = {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)})
32	21	a1i 11	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐴 ⊆ (𝐴 ∪ 𝐵))
33		eqid 2736	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
34		eqid 2736	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) = (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴))
35	10, 25, 12, 33, 34	pwssplit3 21013	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) ∈ (𝐸 LMHom 𝐶))
36	32, 35	syld3an3 1411	. . . . . 6 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) ∈ (𝐸 LMHom 𝐶))
37		fvex 6847	. . . . . . . . 9 ⊢ (0_g‘𝐶) ∈ V
38	34	mptiniseg 6197	. . . . . . . . 9 ⊢ ((0_g‘𝐶) ∈ V → (^◡(𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) “ {(0_g‘𝐶)}) = {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)})
39	37, 38	ax-mp 5	. . . . . . . 8 ⊢ (^◡(𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) “ {(0_g‘𝐶)}) = {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)}
40	39	eqcomi 2745	. . . . . . 7 ⊢ {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)} = (^◡(𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) “ {(0_g‘𝐶)})
41		eqid 2736	. . . . . . 7 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
42		eqid 2736	. . . . . . 7 ⊢ (LSubSp‘𝐸) = (LSubSp‘𝐸)
43	40, 41, 42	lmhmkerlss 21003	. . . . . 6 ⊢ ((𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) ∈ (𝐸 LMHom 𝐶) → {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)} ∈ (LSubSp‘𝐸))
44	36, 43	syl 17	. . . . 5 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)} ∈ (LSubSp‘𝐸))
45	31, 44	eqeltrd 2836	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐾 ∈ (LSubSp‘𝐸))
46		pwssplit4.l	. . . . 5 ⊢ 𝐿 = (𝐸 ↾_s 𝐾)
47	42, 46	reslmhm 21004	. . . 4 ⊢ (((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐸 LMHom 𝐷) ∧ 𝐾 ∈ (LSubSp‘𝐸)) → ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
48	16, 45, 47	syl2anc 584	. . 3 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
49	7, 48	eqeltrid 2840	. 2 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 ∈ (𝐿 LMHom 𝐷))
50	1	fvtresfn 6943	. . . . . . 7 ⊢ (𝑎 ∈ 𝐾 → (𝐹‘𝑎) = (𝑎 ↾ 𝐵))
51		ssexg 5268	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → 𝐵 ∈ V)
52	8, 51	mpan 690	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝑉 → 𝐵 ∈ V)
53	52	3ad2ant2 1134	. . . . . . . . 9 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ V)
54	11, 26	pws0g 18698	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝐵 ∈ V) → (𝐵 × { 0 }) = (0_g‘𝐷))
55	20, 53, 54	syl2anc 584	. . . . . . . 8 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐵 × { 0 }) = (0_g‘𝐷))
56	55	eqcomd 2742	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (0_g‘𝐷) = (𝐵 × { 0 }))
57	50, 56	eqeqan12rd 2751	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝐹‘𝑎) = (0_g‘𝐷) ↔ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })))
58		reseq1 5932	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝑦 ↾ 𝐴) = (𝑎 ↾ 𝐴))
59	58	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })))
60	59, 2	elrab2 3649	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐾 ↔ (𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })))
61		uneq12 4115	. . . . . . . . . . . . 13 ⊢ (((𝑎 ↾ 𝐴) = (𝐴 × { 0 }) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → ((𝑎 ↾ 𝐴) ∪ (𝑎 ↾ 𝐵)) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 })))
62		resundi 5952	. . . . . . . . . . . . 13 ⊢ (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝑎 ↾ 𝐴) ∪ (𝑎 ↾ 𝐵))
63		xpundir 5694	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ 𝐵) × { 0 }) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 }))
64	61, 62, 63	3eqtr4g 2796	. . . . . . . . . . . 12 ⊢ (((𝑎 ↾ 𝐴) = (𝐴 × { 0 }) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) × { 0 }))
65	64	adantll 714	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) × { 0 }))
66	65	adantl 481	. . . . . . . . . 10 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) × { 0 }))
67		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
68		simpl1 1192	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → 𝑅 ∈ LMod)
69		simp2 1137	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ 𝑉)
70	69	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → (𝐴 ∪ 𝐵) ∈ 𝑉)
71		simprll 778	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → 𝑎 ∈ 𝐺)
72	10, 67, 12, 68, 70, 71	pwselbas 17409	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → 𝑎:(𝐴 ∪ 𝐵)⟶(Base‘𝑅))
73		ffn 6662	. . . . . . . . . . 11 ⊢ (𝑎:(𝐴 ∪ 𝐵)⟶(Base‘𝑅) → 𝑎 Fn (𝐴 ∪ 𝐵))
74		fnresdm 6611	. . . . . . . . . . 11 ⊢ (𝑎 Fn (𝐴 ∪ 𝐵) → (𝑎 ↾ (𝐴 ∪ 𝐵)) = 𝑎)
75	72, 73, 74	3syl 18	. . . . . . . . . 10 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → (𝑎 ↾ (𝐴 ∪ 𝐵)) = 𝑎)
76	10, 26	pws0g 18698	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Mnd ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → ((𝐴 ∪ 𝐵) × { 0 }) = (0_g‘𝐸))
77	20, 69, 76	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) × { 0 }) = (0_g‘𝐸))
78	10	pwslmod 20921	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → 𝐸 ∈ LMod)
79	78	3adant3 1132	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐸 ∈ LMod)
80	42	lsssubg 20908	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝐸)) → 𝐾 ∈ (SubGrp‘𝐸))
81	79, 45, 80	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐾 ∈ (SubGrp‘𝐸))
82		eqid 2736	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
83	46, 82	subg0 19062	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (SubGrp‘𝐸) → (0_g‘𝐸) = (0_g‘𝐿))
84	81, 83	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (0_g‘𝐸) = (0_g‘𝐿))
85	77, 84	eqtrd 2771	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) × { 0 }) = (0_g‘𝐿))
86	85	adantr 480	. . . . . . . . . 10 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → ((𝐴 ∪ 𝐵) × { 0 }) = (0_g‘𝐿))
87	66, 75, 86	3eqtr3d 2779	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → 𝑎 = (0_g‘𝐿))
88	87	exp32 420	. . . . . . . 8 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) → ((𝑎 ↾ 𝐵) = (𝐵 × { 0 }) → 𝑎 = (0_g‘𝐿))))
89	60, 88	biimtrid 242	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑎 ∈ 𝐾 → ((𝑎 ↾ 𝐵) = (𝐵 × { 0 }) → 𝑎 = (0_g‘𝐿))))
90	89	imp 406	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝑎 ↾ 𝐵) = (𝐵 × { 0 }) → 𝑎 = (0_g‘𝐿)))
91	57, 90	sylbid 240	. . . . 5 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝐹‘𝑎) = (0_g‘𝐷) → 𝑎 = (0_g‘𝐿)))
92	91	ralrimiva 3128	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑎 ∈ 𝐾 ((𝐹‘𝑎) = (0_g‘𝐷) → 𝑎 = (0_g‘𝐿)))
93		lmghm 20983	. . . . 5 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹 ∈ (𝐿 GrpHom 𝐷))
94	46, 12	ressbas2 17165	. . . . . . 7 ⊢ (𝐾 ⊆ 𝐺 → 𝐾 = (Base‘𝐿))
95	4, 94	ax-mp 5	. . . . . 6 ⊢ 𝐾 = (Base‘𝐿)
96		eqid 2736	. . . . . 6 ⊢ (0_g‘𝐿) = (0_g‘𝐿)
97		eqid 2736	. . . . . 6 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
98	95, 13, 96, 97	ghmf1 19175	. . . . 5 ⊢ (𝐹 ∈ (𝐿 GrpHom 𝐷) → (𝐹:𝐾–1-1→(Base‘𝐷) ↔ ∀𝑎 ∈ 𝐾 ((𝐹‘𝑎) = (0_g‘𝐷) → 𝑎 = (0_g‘𝐿))))
99	49, 93, 98	3syl 18	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐹:𝐾–1-1→(Base‘𝐷) ↔ ∀𝑎 ∈ 𝐾 ((𝐹‘𝑎) = (0_g‘𝐷) → 𝑎 = (0_g‘𝐿))))
100	92, 99	mpbird 257	. . 3 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹:𝐾–1-1→(Base‘𝐷))
101		eqid 2736	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
102	101, 13	lmhmf 20986	. . . . 5 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:(Base‘𝐿)⟶(Base‘𝐷))
103		frn 6669	. . . . 5 ⊢ (𝐹:(Base‘𝐿)⟶(Base‘𝐷) → ran 𝐹 ⊆ (Base‘𝐷))
104	49, 102, 103	3syl 18	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ran 𝐹 ⊆ (Base‘𝐷))
105		reseq1 5932	. . . . . . 7 ⊢ (𝑥 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑥 ↾ 𝐵) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
106	11, 67, 13	pwselbasb 17408	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ 𝐵 ∈ V) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
107	17, 53, 106	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
108	107	biimpa 476	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎:𝐵⟶(Base‘𝑅))
109	26	fvexi 6848	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
110	109	fconst 6720	. . . . . . . . . . . . 13 ⊢ (𝐴 × { 0 }):𝐴⟶{ 0 }
111	110	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶{ 0 })
112	20	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑅 ∈ Mnd)
113	67, 26	mndidcl 18674	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅))
114	112, 113	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 0 ∈ (Base‘𝑅))
115	114	snssd 4765	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → { 0 } ⊆ (Base‘𝑅))
116	111, 115	fssd 6679	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶(Base‘𝑅))
117		incom 4161	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
118		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
119	117, 118	eqtrid 2783	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐵 ∩ 𝐴) = ∅)
120	119	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵 ∩ 𝐴) = ∅)
121		fun 6696	. . . . . . . . . . 11 ⊢ (((𝑎:𝐵⟶(Base‘𝑅) ∧ (𝐴 × { 0 }):𝐴⟶(Base‘𝑅)) ∧ (𝐵 ∩ 𝐴) = ∅) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵 ∪ 𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
122	108, 116, 120, 121	syl21anc 837	. . . . . . . . . 10 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵 ∪ 𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
123		uncom 4110	. . . . . . . . . . 11 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
124		unidm 4109	. . . . . . . . . . 11 ⊢ ((Base‘𝑅) ∪ (Base‘𝑅)) = (Base‘𝑅)
125	123, 124	feq23i 6656	. . . . . . . . . 10 ⊢ ((𝑎 ∪ (𝐴 × { 0 })):(𝐵 ∪ 𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)) ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅))
126	122, 125	sylib 218	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅))
127	10, 67, 12	pwselbasb 17408	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅)))
128	127	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅)))
129	128	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅)))
130	126, 129	mpbird 257	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺)
131		simpl3 1194	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 ∩ 𝐵) = ∅)
132	117, 131	eqtrid 2783	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵 ∩ 𝐴) = ∅)
133		ffn 6662	. . . . . . . . . . . 12 ⊢ (𝑎:𝐵⟶(Base‘𝑅) → 𝑎 Fn 𝐵)
134		fnresdisj 6612	. . . . . . . . . . . 12 ⊢ (𝑎 Fn 𝐵 → ((𝐵 ∩ 𝐴) = ∅ ↔ (𝑎 ↾ 𝐴) = ∅))
135	108, 133, 134	3syl 18	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐵 ∩ 𝐴) = ∅ ↔ (𝑎 ↾ 𝐴) = ∅))
136	132, 135	mpbid 232	. . . . . . . . . 10 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ↾ 𝐴) = ∅)
137		fnconstg 6722	. . . . . . . . . . . 12 ⊢ ( 0 ∈ V → (𝐴 × { 0 }) Fn 𝐴)
138		fnresdm 6611	. . . . . . . . . . . 12 ⊢ ((𝐴 × { 0 }) Fn 𝐴 → ((𝐴 × { 0 }) ↾ 𝐴) = (𝐴 × { 0 }))
139	109, 137, 138	mp2b 10	. . . . . . . . . . 11 ⊢ ((𝐴 × { 0 }) ↾ 𝐴) = (𝐴 × { 0 })
140	139	a1i 11	. . . . . . . . . 10 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐴 × { 0 }) ↾ 𝐴) = (𝐴 × { 0 }))
141	136, 140	uneq12d 4121	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴)) = (∅ ∪ (𝐴 × { 0 })))
142		resundir 5953	. . . . . . . . 9 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = ((𝑎 ↾ 𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴))
143		uncom 4110	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐴 × { 0 })) = ((𝐴 × { 0 }) ∪ ∅)
144		un0 4346	. . . . . . . . . 10 ⊢ ((𝐴 × { 0 }) ∪ ∅) = (𝐴 × { 0 })
145	143, 144	eqtr2i 2760	. . . . . . . . 9 ⊢ (𝐴 × { 0 }) = (∅ ∪ (𝐴 × { 0 }))
146	141, 142, 145	3eqtr4g 2796	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 }))
147		reseq1 5932	. . . . . . . . . 10 ⊢ (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑦 ↾ 𝐴) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴))
148	147	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
149	148, 2	elrab2 3649	. . . . . . . 8 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 ↔ ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ∧ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
150	130, 146, 149	sylanbrc 583	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾)
151	130	resexd 5987	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
152	1, 105, 150, 151	fvmptd3 6964	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
153		resundir 5953	. . . . . . 7 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵))
154		fnresdm 6611	. . . . . . . . . 10 ⊢ (𝑎 Fn 𝐵 → (𝑎 ↾ 𝐵) = 𝑎)
155	108, 133, 154	3syl 18	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ↾ 𝐵) = 𝑎)
156		ffn 6662	. . . . . . . . . . . . 13 ⊢ ((𝐴 × { 0 }):𝐴⟶{ 0 } → (𝐴 × { 0 }) Fn 𝐴)
157		fnresdisj 6612	. . . . . . . . . . . . 13 ⊢ ((𝐴 × { 0 }) Fn 𝐴 → ((𝐴 ∩ 𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅))
158	110, 156, 157	mp2b 10	. . . . . . . . . . . 12 ⊢ ((𝐴 ∩ 𝐵) = ∅ ↔ ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
159	158	biimpi 216	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
160	159	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
161	160	adantr 480	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
162	155, 161	uneq12d 4121	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = (𝑎 ∪ ∅))
163		un0 4346	. . . . . . . 8 ⊢ (𝑎 ∪ ∅) = 𝑎
164	162, 163	eqtrdi 2787	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = 𝑎)
165	153, 164	eqtrid 2783	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = 𝑎)
166	152, 165	eqtrd 2771	. . . . 5 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = 𝑎)
167	95, 13	lmhmf 20986	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:𝐾⟶(Base‘𝐷))
168		ffn 6662	. . . . . . . 8 ⊢ (𝐹:𝐾⟶(Base‘𝐷) → 𝐹 Fn 𝐾)
169	49, 167, 168	3syl 18	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 Fn 𝐾)
170	169	adantr 480	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝐹 Fn 𝐾)
171		fnfvelrn 7025	. . . . . 6 ⊢ ((𝐹 Fn 𝐾 ∧ (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
172	170, 150, 171	syl2anc 584	. . . . 5 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
173	166, 172	eqeltrrd 2837	. . . 4 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎 ∈ ran 𝐹)
174	104, 173	eqelssd 3955	. . 3 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ran 𝐹 = (Base‘𝐷))
175		dff1o5 6783	. . 3 ⊢ (𝐹:𝐾–1-1-onto→(Base‘𝐷) ↔ (𝐹:𝐾–1-1→(Base‘𝐷) ∧ ran 𝐹 = (Base‘𝐷)))
176	100, 174, 175	sylanbrc 583	. 2 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹:𝐾–1-1-onto→(Base‘𝐷))
177	95, 13	islmim 21014	. 2 ⊢ (𝐹 ∈ (𝐿 LMIso 𝐷) ↔ (𝐹 ∈ (𝐿 LMHom 𝐷) ∧ 𝐹:𝐾–1-1-onto→(Base‘𝐷)))
178	49, 176, 177	sylanbrc 583	1 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ∀wral 3051 {crab 3399 Vcvv 3440 ∪ cun 3899 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 {csn 4580 ↦ cmpt 5179 × cxp 5622 ^◡ccnv 5623 ran crn 5625 ↾ cres 5626 “ cima 5627 Fn wfn 6487 ⟶wf 6488 –1-1→wf1 6489 –1-1-onto→wf1o 6491 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 ↾_s cress 17157 0_gc0g 17359 ↑_s cpws 17366 Mndcmnd 18659 Grpcgrp 18863 SubGrpcsubg 19050 GrpHom cghm 19141 LModclmod 20811 LSubSpclss 20882 LMHom clmhm 20971 LMIso clmim 20972

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 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-iun 4948 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-er 8635 df-map 8765 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-fz 13424 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-hom 17201 df-cco 17202 df-0g 17361 df-prds 17367 df-pws 17369 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18866 df-minusg 18867 df-sbg 18868 df-subg 19053 df-ghm 19142 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-lmod 20813 df-lss 20883 df-lmhm 20974 df-lmim 20975

This theorem is referenced by: pwslnmlem2 43335

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