pwssplit4 - Mathbox for Stefan O'Rear

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Theorem pwssplit4 40914

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 4013	. . . . . 6 ⊢ {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })} ⊆ 𝐺
4	2, 3	eqsstri 3955	. . . . 5 ⊢ 𝐾 ⊆ 𝐺
5		resmpt 5945	. . . . 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 2769	. . 3 ⊢ 𝐹 = ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾)
8		ssun2 4107	. . . . . 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 2738	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
14		eqid 2738	. . . . . 6 ⊢ (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) = (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵))
15	10, 11, 12, 13, 14	pwssplit3 20323	. . . . 5 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐸 LMHom 𝐷))
16	9, 15	syld3an3 1408	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐸 LMHom 𝐷))
17		simp1 1135	. . . . . . . . . 10 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 ∈ LMod)
18		lmodgrp 20130	. . . . . . . . . 10 ⊢ (𝑅 ∈ LMod → 𝑅 ∈ Grp)
19		grpmnd 18584	. . . . . . . . . 10 ⊢ (𝑅 ∈ Grp → 𝑅 ∈ Mnd)
20	17, 18, 19	3syl 18	. . . . . . . . 9 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 ∈ Mnd)
21		ssun1 4106	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
22		ssexg 5247	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
23	21, 22	mpan 687	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝑉 → 𝐴 ∈ V)
24	23	3ad2ant2 1133	. . . . . . . . 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 18421	. . . . . . . . 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 2749	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ (𝑦 ↾ 𝐴) = (0_g‘𝐶)))
30	29	rabbidv 3414	. . . . . 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 2790	. . . . 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 2738	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
34		eqid 2738	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) = (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴))
35	10, 25, 12, 33, 34	pwssplit3 20323	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) ∈ (𝐸 LMHom 𝐶))
36	32, 35	syld3an3 1408	. . . . . 6 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) ∈ (𝐸 LMHom 𝐶))
37		fvex 6787	. . . . . . . . 9 ⊢ (0_g‘𝐶) ∈ V
38	34	mptiniseg 6142	. . . . . . . . 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 2747	. . . . . . 7 ⊢ {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)} = (^◡(𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) “ {(0_g‘𝐶)})
41		eqid 2738	. . . . . . 7 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
42		eqid 2738	. . . . . . 7 ⊢ (LSubSp‘𝐸) = (LSubSp‘𝐸)
43	40, 41, 42	lmhmkerlss 20313	. . . . . 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 2839	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐾 ∈ (LSubSp‘𝐸))
46		pwssplit4.l	. . . . 5 ⊢ 𝐿 = (𝐸 ↾_s 𝐾)
47	42, 46	reslmhm 20314	. . . 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 2843	. 2 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 ∈ (𝐿 LMHom 𝐷))
50	1	fvtresfn 6877	. . . . . . 7 ⊢ (𝑎 ∈ 𝐾 → (𝐹‘𝑎) = (𝑎 ↾ 𝐵))
51		ssexg 5247	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → 𝐵 ∈ V)
52	8, 51	mpan 687	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝑉 → 𝐵 ∈ V)
53	52	3ad2ant2 1133	. . . . . . . . 9 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ V)
54	11, 26	pws0g 18421	. . . . . . . . 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 2744	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (0_g‘𝐷) = (𝐵 × { 0 }))
57	50, 56	eqeqan12rd 2753	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝐹‘𝑎) = (0_g‘𝐷) ↔ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })))
58		reseq1 5885	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝑦 ↾ 𝐴) = (𝑎 ↾ 𝐴))
59	58	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })))
60	59, 2	elrab2 3627	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐾 ↔ (𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })))
61		uneq12 4092	. . . . . . . . . . . . 13 ⊢ (((𝑎 ↾ 𝐴) = (𝐴 × { 0 }) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → ((𝑎 ↾ 𝐴) ∪ (𝑎 ↾ 𝐵)) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 })))
62		resundi 5905	. . . . . . . . . . . . 13 ⊢ (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝑎 ↾ 𝐴) ∪ (𝑎 ↾ 𝐵))
63		xpundir 5656	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ 𝐵) × { 0 }) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 }))
64	61, 62, 63	3eqtr4g 2803	. . . . . . . . . . . 12 ⊢ (((𝑎 ↾ 𝐴) = (𝐴 × { 0 }) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) × { 0 }))
65	64	adantll 711	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) × { 0 }))
66	65	adantl 482	. . . . . . . . . 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 2738	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
68		simpl1 1190	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → 𝑅 ∈ LMod)
69		simp2 1136	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ 𝑉)
70	69	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → (𝐴 ∪ 𝐵) ∈ 𝑉)
71		simprll 776	. . . . . . . . . . . 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 17200	. . . . . . . . . . 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 6600	. . . . . . . . . . 11 ⊢ (𝑎:(𝐴 ∪ 𝐵)⟶(Base‘𝑅) → 𝑎 Fn (𝐴 ∪ 𝐵))
74		fnresdm 6551	. . . . . . . . . . 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 18421	. . . . . . . . . . . . 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 20232	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → 𝐸 ∈ LMod)
79	78	3adant3 1131	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐸 ∈ LMod)
80	42	lsssubg 20219	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
83	46, 82	subg0 18761	. . . . . . . . . . . . 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 2778	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) × { 0 }) = (0_g‘𝐿))
86	85	adantr 481	. . . . . . . . . 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 2786	. . . . . . . . 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 421	. . . . . . . 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	syl5bi 241	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑎 ∈ 𝐾 → ((𝑎 ↾ 𝐵) = (𝐵 × { 0 }) → 𝑎 = (0_g‘𝐿))))
90	89	imp 407	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝑎 ↾ 𝐵) = (𝐵 × { 0 }) → 𝑎 = (0_g‘𝐿)))
91	57, 90	sylbid 239	. . . . 5 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝐹‘𝑎) = (0_g‘𝐷) → 𝑎 = (0_g‘𝐿)))
92	91	ralrimiva 3103	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑎 ∈ 𝐾 ((𝐹‘𝑎) = (0_g‘𝐷) → 𝑎 = (0_g‘𝐿)))
93		lmghm 20293	. . . . 5 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹 ∈ (𝐿 GrpHom 𝐷))
94	46, 12	ressbas2 16949	. . . . . . 7 ⊢ (𝐾 ⊆ 𝐺 → 𝐾 = (Base‘𝐿))
95	4, 94	ax-mp 5	. . . . . 6 ⊢ 𝐾 = (Base‘𝐿)
96		eqid 2738	. . . . . 6 ⊢ (0_g‘𝐿) = (0_g‘𝐿)
97		eqid 2738	. . . . . 6 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
98	95, 13, 96, 97	ghmf1 18863	. . . . 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 256	. . 3 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹:𝐾–1-1→(Base‘𝐷))
101		eqid 2738	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
102	101, 13	lmhmf 20296	. . . . 5 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:(Base‘𝐿)⟶(Base‘𝐷))
103		frn 6607	. . . . 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 5885	. . . . . . 7 ⊢ (𝑥 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑥 ↾ 𝐵) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
106	11, 67, 13	pwselbasb 17199	. . . . . . . . . . . . 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 477	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎:𝐵⟶(Base‘𝑅))
109	26	fvexi 6788	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
110	109	fconst 6660	. . . . . . . . . . . . 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 481	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑅 ∈ Mnd)
113	67, 26	mndidcl 18400	. . . . . . . . . . . . . 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 4742	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → { 0 } ⊆ (Base‘𝑅))
116	111, 115	fssd 6618	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶(Base‘𝑅))
117		incom 4135	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
118		simp3 1137	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
119	117, 118	eqtrid 2790	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐵 ∩ 𝐴) = ∅)
120	119	adantr 481	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵 ∩ 𝐴) = ∅)
121		fun 6636	. . . . . . . . . . 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 835	. . . . . . . . . 10 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵 ∪ 𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
123		uncom 4087	. . . . . . . . . . 11 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
124		unidm 4086	. . . . . . . . . . 11 ⊢ ((Base‘𝑅) ∪ (Base‘𝑅)) = (Base‘𝑅)
125	123, 124	feq23i 6594	. . . . . . . . . 10 ⊢ ((𝑎 ∪ (𝐴 × { 0 })):(𝐵 ∪ 𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)) ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅))
126	122, 125	sylib 217	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅))
127	10, 67, 12	pwselbasb 17199	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅)))
128	127	3adant3 1131	. . . . . . . . . 10 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅)))
129	128	adantr 481	. . . . . . . . 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 256	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺)
131		simpl3 1192	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 ∩ 𝐵) = ∅)
132	117, 131	eqtrid 2790	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵 ∩ 𝐴) = ∅)
133		ffn 6600	. . . . . . . . . . . 12 ⊢ (𝑎:𝐵⟶(Base‘𝑅) → 𝑎 Fn 𝐵)
134		fnresdisj 6552	. . . . . . . . . . . 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 231	. . . . . . . . . 10 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ↾ 𝐴) = ∅)
137		fnconstg 6662	. . . . . . . . . . . 12 ⊢ ( 0 ∈ V → (𝐴 × { 0 }) Fn 𝐴)
138		fnresdm 6551	. . . . . . . . . . . 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 4098	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴)) = (∅ ∪ (𝐴 × { 0 })))
142		resundir 5906	. . . . . . . . 9 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = ((𝑎 ↾ 𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴))
143		uncom 4087	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐴 × { 0 })) = ((𝐴 × { 0 }) ∪ ∅)
144		un0 4324	. . . . . . . . . 10 ⊢ ((𝐴 × { 0 }) ∪ ∅) = (𝐴 × { 0 })
145	143, 144	eqtr2i 2767	. . . . . . . . 9 ⊢ (𝐴 × { 0 }) = (∅ ∪ (𝐴 × { 0 }))
146	141, 142, 145	3eqtr4g 2803	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 }))
147		reseq1 5885	. . . . . . . . . 10 ⊢ (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑦 ↾ 𝐴) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴))
148	147	eqeq1d 2740	. . . . . . . . 9 ⊢ (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
149	148, 2	elrab2 3627	. . . . . . . 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 5938	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
152	1, 105, 150, 151	fvmptd3 6898	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
153		resundir 5906	. . . . . . 7 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵))
154		fnresdm 6551	. . . . . . . . . 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 6600	. . . . . . . . . . . . 13 ⊢ ((𝐴 × { 0 }):𝐴⟶{ 0 } → (𝐴 × { 0 }) Fn 𝐴)
157		fnresdisj 6552	. . . . . . . . . . . . 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 215	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
160	159	3ad2ant3 1134	. . . . . . . . . 10 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
161	160	adantr 481	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
162	155, 161	uneq12d 4098	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = (𝑎 ∪ ∅))
163		un0 4324	. . . . . . . 8 ⊢ (𝑎 ∪ ∅) = 𝑎
164	162, 163	eqtrdi 2794	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = 𝑎)
165	153, 164	eqtrid 2790	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = 𝑎)
166	152, 165	eqtrd 2778	. . . . 5 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = 𝑎)
167	95, 13	lmhmf 20296	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:𝐾⟶(Base‘𝐷))
168		ffn 6600	. . . . . . . 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 481	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝐹 Fn 𝐾)
171		fnfvelrn 6958	. . . . . 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 2840	. . . 4 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎 ∈ ran 𝐹)
174	104, 173	eqelssd 3942	. . 3 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ran 𝐹 = (Base‘𝐷))
175		dff1o5 6725	. . 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 20324	. 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 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 {crab 3068 Vcvv 3432 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 {csn 4561 ↦ cmpt 5157 × cxp 5587 ^◡ccnv 5588 ran crn 5590 ↾ cres 5591 “ cima 5592 Fn wfn 6428 ⟶wf 6429 –1-1→wf1 6430 –1-1-onto→wf1o 6432 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 ↾_s cress 16941 0_gc0g 17150 ↑_s cpws 17157 Mndcmnd 18385 Grpcgrp 18577 SubGrpcsubg 18749 GrpHom cghm 18831 LModclmod 20123 LSubSpclss 20193 LMHom clmhm 20281 LMIso clmim 20282

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-0g 17152 df-prds 17158 df-pws 17160 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-ghm 18832 df-mgp 19721 df-ur 19738 df-ring 19785 df-lmod 20125 df-lss 20194 df-lmhm 20284 df-lmim 20285

This theorem is referenced by: pwslnmlem2 40918

W3C validator