pwssplit4 - Mathbox for Stefan O'Rear

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Theorem pwssplit4 43060

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 4055	. . . . . 6 ⊢ {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })} ⊆ 𝐺
4	2, 3	eqsstri 4005	. . . . 5 ⊢ 𝐾 ⊆ 𝐺
5		resmpt 6024	. . . . 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 2761	. . 3 ⊢ 𝐹 = ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾)
8		ssun2 4154	. . . . . 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 2735	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
14		eqid 2735	. . . . . 6 ⊢ (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) = (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵))
15	10, 11, 12, 13, 14	pwssplit3 21017	. . . . 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 20822	. . . . . . . . . 10 ⊢ (𝑅 ∈ LMod → 𝑅 ∈ Grp)
19		grpmnd 18921	. . . . . . . . . 10 ⊢ (𝑅 ∈ Grp → 𝑅 ∈ Mnd)
20	17, 18, 19	3syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 ∈ Mnd)
21		ssun1 4153	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
22		ssexg 5293	. . . . . . . . . . 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 18749	. . . . . . . . 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 2746	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ (𝑦 ↾ 𝐴) = (0_g‘𝐶)))
30	29	rabbidv 3423	. . . . . 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 2782	. . . . 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 2735	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
34		eqid 2735	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) = (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴))
35	10, 25, 12, 33, 34	pwssplit3 21017	. . . . . . 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 6888	. . . . . . . . 9 ⊢ (0_g‘𝐶) ∈ V
38	34	mptiniseg 6228	. . . . . . . . 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 2744	. . . . . . 7 ⊢ {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)} = (^◡(𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) “ {(0_g‘𝐶)})
41		eqid 2735	. . . . . . 7 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
42		eqid 2735	. . . . . . 7 ⊢ (LSubSp‘𝐸) = (LSubSp‘𝐸)
43	40, 41, 42	lmhmkerlss 21007	. . . . . 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 2834	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐾 ∈ (LSubSp‘𝐸))
46		pwssplit4.l	. . . . 5 ⊢ 𝐿 = (𝐸 ↾_s 𝐾)
47	42, 46	reslmhm 21008	. . . 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 2838	. 2 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 ∈ (𝐿 LMHom 𝐷))
50	1	fvtresfn 6987	. . . . . . 7 ⊢ (𝑎 ∈ 𝐾 → (𝐹‘𝑎) = (𝑎 ↾ 𝐵))
51		ssexg 5293	. . . . . . . . . . 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 18749	. . . . . . . . 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 2741	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (0_g‘𝐷) = (𝐵 × { 0 }))
57	50, 56	eqeqan12rd 2750	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝐹‘𝑎) = (0_g‘𝐷) ↔ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })))
58		reseq1 5960	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝑦 ↾ 𝐴) = (𝑎 ↾ 𝐴))
59	58	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })))
60	59, 2	elrab2 3674	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐾 ↔ (𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })))
61		uneq12 4138	. . . . . . . . . . . . 13 ⊢ (((𝑎 ↾ 𝐴) = (𝐴 × { 0 }) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → ((𝑎 ↾ 𝐴) ∪ (𝑎 ↾ 𝐵)) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 })))
62		resundi 5980	. . . . . . . . . . . . 13 ⊢ (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝑎 ↾ 𝐴) ∪ (𝑎 ↾ 𝐵))
63		xpundir 5724	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ 𝐵) × { 0 }) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 }))
64	61, 62, 63	3eqtr4g 2795	. . . . . . . . . . . 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 2735	. . . . . . . . . . . 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 17501	. . . . . . . . . . 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 6705	. . . . . . . . . . 11 ⊢ (𝑎:(𝐴 ∪ 𝐵)⟶(Base‘𝑅) → 𝑎 Fn (𝐴 ∪ 𝐵))
74		fnresdm 6656	. . . . . . . . . . 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 18749	. . . . . . . . . . . . 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 20925	. . . . . . . . . . . . . . 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 20912	. . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
83	46, 82	subg0 19113	. . . . . . . . . . . . 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 2770	. . . . . . . . . . 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 2778	. . . . . . . . 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 3132	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑎 ∈ 𝐾 ((𝐹‘𝑎) = (0_g‘𝐷) → 𝑎 = (0_g‘𝐿)))
93		lmghm 20987	. . . . 5 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹 ∈ (𝐿 GrpHom 𝐷))
94	46, 12	ressbas2 17257	. . . . . . 7 ⊢ (𝐾 ⊆ 𝐺 → 𝐾 = (Base‘𝐿))
95	4, 94	ax-mp 5	. . . . . 6 ⊢ 𝐾 = (Base‘𝐿)
96		eqid 2735	. . . . . 6 ⊢ (0_g‘𝐿) = (0_g‘𝐿)
97		eqid 2735	. . . . . 6 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
98	95, 13, 96, 97	ghmf1 19227	. . . . 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 2735	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
102	101, 13	lmhmf 20990	. . . . 5 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:(Base‘𝐿)⟶(Base‘𝐷))
103		frn 6712	. . . . 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 5960	. . . . . . 7 ⊢ (𝑥 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑥 ↾ 𝐵) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
106	11, 67, 13	pwselbasb 17500	. . . . . . . . . . . . 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 6889	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
110	109	fconst 6763	. . . . . . . . . . . . 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 18725	. . . . . . . . . . . . . 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 4785	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → { 0 } ⊆ (Base‘𝑅))
116	111, 115	fssd 6722	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶(Base‘𝑅))
117		incom 4184	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
118		simp3 1138	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
119	117, 118	eqtrid 2782	. . . . . . . . . . . 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 6739	. . . . . . . . . . 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 4133	. . . . . . . . . . 11 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
124		unidm 4132	. . . . . . . . . . 11 ⊢ ((Base‘𝑅) ∪ (Base‘𝑅)) = (Base‘𝑅)
125	123, 124	feq23i 6699	. . . . . . . . . 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 17500	. . . . . . . . . . 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 2782	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵 ∩ 𝐴) = ∅)
133		ffn 6705	. . . . . . . . . . . 12 ⊢ (𝑎:𝐵⟶(Base‘𝑅) → 𝑎 Fn 𝐵)
134		fnresdisj 6657	. . . . . . . . . . . 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 6765	. . . . . . . . . . . 12 ⊢ ( 0 ∈ V → (𝐴 × { 0 }) Fn 𝐴)
138		fnresdm 6656	. . . . . . . . . . . 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 4144	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴)) = (∅ ∪ (𝐴 × { 0 })))
142		resundir 5981	. . . . . . . . 9 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = ((𝑎 ↾ 𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴))
143		uncom 4133	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐴 × { 0 })) = ((𝐴 × { 0 }) ∪ ∅)
144		un0 4369	. . . . . . . . . 10 ⊢ ((𝐴 × { 0 }) ∪ ∅) = (𝐴 × { 0 })
145	143, 144	eqtr2i 2759	. . . . . . . . 9 ⊢ (𝐴 × { 0 }) = (∅ ∪ (𝐴 × { 0 }))
146	141, 142, 145	3eqtr4g 2795	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 }))
147		reseq1 5960	. . . . . . . . . 10 ⊢ (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑦 ↾ 𝐴) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴))
148	147	eqeq1d 2737	. . . . . . . . 9 ⊢ (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
149	148, 2	elrab2 3674	. . . . . . . 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 6015	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
152	1, 105, 150, 151	fvmptd3 7008	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
153		resundir 5981	. . . . . . 7 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵))
154		fnresdm 6656	. . . . . . . . . 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 6705	. . . . . . . . . . . . 13 ⊢ ((𝐴 × { 0 }):𝐴⟶{ 0 } → (𝐴 × { 0 }) Fn 𝐴)
157		fnresdisj 6657	. . . . . . . . . . . . 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 4144	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = (𝑎 ∪ ∅))
163		un0 4369	. . . . . . . 8 ⊢ (𝑎 ∪ ∅) = 𝑎
164	162, 163	eqtrdi 2786	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = 𝑎)
165	153, 164	eqtrid 2782	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = 𝑎)
166	152, 165	eqtrd 2770	. . . . 5 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = 𝑎)
167	95, 13	lmhmf 20990	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:𝐾⟶(Base‘𝐷))
168		ffn 6705	. . . . . . . 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 7069	. . . . . 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 2835	. . . 4 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎 ∈ ran 𝐹)
174	104, 173	eqelssd 3980	. . 3 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ran 𝐹 = (Base‘𝐷))
175		dff1o5 6826	. . 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 21018	. 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 1540 ∈ wcel 2108 ∀wral 3051 {crab 3415 Vcvv 3459 ∪ cun 3924 ∩ cin 3925 ⊆ wss 3926 ∅c0 4308 {csn 4601 ↦ cmpt 5201 × cxp 5652 ^◡ccnv 5653 ran crn 5655 ↾ cres 5656 “ cima 5657 Fn wfn 6525 ⟶wf 6526 –1-1→wf1 6527 –1-1-onto→wf1o 6529 ‘cfv 6530 (class class class)co 7403 Basecbs 17226 ↾_s cress 17249 0_gc0g 17451 ↑_s cpws 17458 Mndcmnd 18710 Grpcgrp 18914 SubGrpcsubg 19101 GrpHom cghm 19193 LModclmod 20815 LSubSpclss 20886 LMHom clmhm 20975 LMIso clmim 20976

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 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8717 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-sup 9452 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-fz 13523 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-hom 17293 df-cco 17294 df-0g 17453 df-prds 17459 df-pws 17461 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-grp 18917 df-minusg 18918 df-sbg 18919 df-subg 19104 df-ghm 19194 df-cmn 19761 df-abl 19762 df-mgp 20099 df-rng 20111 df-ur 20140 df-ring 20193 df-lmod 20817 df-lss 20887 df-lmhm 20978 df-lmim 20979

This theorem is referenced by: pwslnmlem2 43064

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