pwssplit4 - Mathbox for Stefan O'Rear

	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 43849

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 4035	. . . . . 6 ⊢ {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })} ⊆ 𝐺
4	2, 3	eqsstri 3984	. . . . 5 ⊢ 𝐾 ⊆ 𝐺
5		resmpt 6041	. . . . 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 2791	. . 3 ⊢ 𝐹 = ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾)
8		ssun2 4132	. . . . . 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 2765	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
14		eqid 2765	. . . . . 6 ⊢ (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) = (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵))
15	10, 11, 12, 13, 14	pwssplit3 21212	. . . . 5 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ 𝐵 ⊆ (𝐴 ∪ 𝐵)) → (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐸 LMHom 𝐷))
16	9, 15	syld3an3 1436	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐸 LMHom 𝐷))
17		simp1 1154	. . . . . . . . . 10 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 ∈ LMod)
18		lmodgrp 21018	. . . . . . . . . 10 ⊢ (𝑅 ∈ LMod → 𝑅 ∈ Grp)
19		grpmnd 19031	. . . . . . . . . 10 ⊢ (𝑅 ∈ Grp → 𝑅 ∈ Mnd)
20	17, 18, 19	3syl 19	. . . . . . . . 9 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 ∈ Mnd)
21		ssun1 4131	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
22		ssexg 5292	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → 𝐴 ∈ V)
23	21, 22	mpan 703	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝑉 → 𝐴 ∈ V)
24	23	3ad2ant2 1152	. . . . . . . . 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 18855	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝐴 ∈ V) → (𝐴 × { 0 }) = (0_g‘𝐶))
28	20, 24, 27	syl2anc 596	. . . . . . . 8 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 × { 0 }) = (0_g‘𝐶))
29	28	eqeq2d 2776	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ (𝑦 ↾ 𝐴) = (0_g‘𝐶)))
30	29	rabbidv 3425	. . . . . 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 2812	. . . . 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 2765	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
34		eqid 2765	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) = (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴))
35	10, 25, 12, 33, 34	pwssplit3 21212	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ 𝐴 ⊆ (𝐴 ∪ 𝐵)) → (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) ∈ (𝐸 LMHom 𝐶))
36	32, 35	syld3an3 1436	. . . . . 6 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) ∈ (𝐸 LMHom 𝐶))
37		fvex 6898	. . . . . . . . 9 ⊢ (0_g‘𝐶) ∈ V
38	34	mptiniseg 6242	. . . . . . . . 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 2774	. . . . . . 7 ⊢ {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)} = (^◡(𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) “ {(0_g‘𝐶)})
41		eqid 2765	. . . . . . 7 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
42		eqid 2765	. . . . . . 7 ⊢ (LSubSp‘𝐸) = (LSubSp‘𝐸)
43	40, 41, 42	lmhmkerlss 21202	. . . . . 6 ⊢ ((𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) ∈ (𝐸 LMHom 𝐶) → {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)} ∈ (LSubSp‘𝐸))
44	36, 43	syl 18	. . . . 5 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)} ∈ (LSubSp‘𝐸))
45	31, 44	eqeltrd 2865	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐾 ∈ (LSubSp‘𝐸))
46		pwssplit4.l	. . . . 5 ⊢ 𝐿 = (𝐸 ↾_s 𝐾)
47	42, 46	reslmhm 21203	. . . 4 ⊢ (((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ∈ (𝐸 LMHom 𝐷) ∧ 𝐾 ∈ (LSubSp‘𝐸)) → ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
48	16, 45, 47	syl2anc 596	. . 3 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾) ∈ (𝐿 LMHom 𝐷))
49	7, 48	eqeltrid 2869	. 2 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 ∈ (𝐿 LMHom 𝐷))
50	1	fvtresfn 6996	. . . . . . 7 ⊢ (𝑎 ∈ 𝐾 → (𝐹‘𝑎) = (𝑎 ↾ 𝐵))
51		ssexg 5292	. . . . . . . . . . 11 ⊢ ((𝐵 ⊆ (𝐴 ∪ 𝐵) ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → 𝐵 ∈ V)
52	8, 51	mpan 703	. . . . . . . . . 10 ⊢ ((𝐴 ∪ 𝐵) ∈ 𝑉 → 𝐵 ∈ V)
53	52	3ad2ant2 1152	. . . . . . . . 9 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐵 ∈ V)
54	11, 26	pws0g 18855	. . . . . . . . 9 ⊢ ((𝑅 ∈ Mnd ∧ 𝐵 ∈ V) → (𝐵 × { 0 }) = (0_g‘𝐷))
55	20, 53, 54	syl2anc 596	. . . . . . . 8 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐵 × { 0 }) = (0_g‘𝐷))
56	55	eqcomd 2771	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (0_g‘𝐷) = (𝐵 × { 0 }))
57	50, 56	eqeqan12rd 2780	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝐹‘𝑎) = (0_g‘𝐷) ↔ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })))
58		reseq1 5974	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝑦 ↾ 𝐴) = (𝑎 ↾ 𝐴))
59	58	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑦 = 𝑎 → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })))
60	59, 2	elrab2 3656	. . . . . . . 8 ⊢ (𝑎 ∈ 𝐾 ↔ (𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })))
61		uneq12 4117	. . . . . . . . . . . . 13 ⊢ (((𝑎 ↾ 𝐴) = (𝐴 × { 0 }) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → ((𝑎 ↾ 𝐴) ∪ (𝑎 ↾ 𝐵)) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 })))
62		resundi 5994	. . . . . . . . . . . . 13 ⊢ (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝑎 ↾ 𝐴) ∪ (𝑎 ↾ 𝐵))
63		xpundir 5733	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ 𝐵) × { 0 }) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 }))
64	61, 62, 63	3eqtr4g 2825	. . . . . . . . . . . 12 ⊢ (((𝑎 ↾ 𝐴) = (𝐴 × { 0 }) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) × { 0 }))
65	64	adantll 727	. . . . . . . . . . 11 ⊢ (((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝐴 ∪ 𝐵) × { 0 }))
66	65	adantl 487	. . . . . . . . . 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 2765	. . . . . . . . . . . 12 ⊢ (Base‘𝑅) = (Base‘𝑅)
68		simpl1 1210	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → 𝑅 ∈ LMod)
69		simp2 1155	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∪ 𝐵) ∈ 𝑉)
70	69	adantr 486	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ ((𝑎 ∈ 𝐺 ∧ (𝑎 ↾ 𝐴) = (𝐴 × { 0 })) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 }))) → (𝐴 ∪ 𝐵) ∈ 𝑉)
71		simprll 791	. . . . . . . . . . . 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 17560	. . . . . . . . . . 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 6709	. . . . . . . . . . 11 ⊢ (𝑎:(𝐴 ∪ 𝐵)⟶(Base‘𝑅) → 𝑎 Fn (𝐴 ∪ 𝐵))
74		fnresdm 6658	. . . . . . . . . . 11 ⊢ (𝑎 Fn (𝐴 ∪ 𝐵) → (𝑎 ↾ (𝐴 ∪ 𝐵)) = 𝑎)
75	72, 73, 74	3syl 19	. . . . . . . . . 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 18855	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ Mnd ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → ((𝐴 ∪ 𝐵) × { 0 }) = (0_g‘𝐸))
77	20, 69, 76	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) × { 0 }) = (0_g‘𝐸))
78	10	pwslmod 21121	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → 𝐸 ∈ LMod)
79	78	3adant3 1150	. . . . . . . . . . . . . 14 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐸 ∈ LMod)
80	42	lsssubg 21108	. . . . . . . . . . . . . 14 ⊢ ((𝐸 ∈ LMod ∧ 𝐾 ∈ (LSubSp‘𝐸)) → 𝐾 ∈ (SubGrp‘𝐸))
81	79, 45, 80	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐾 ∈ (SubGrp‘𝐸))
82		eqid 2765	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
83	46, 82	subg0 19222	. . . . . . . . . . . . 13 ⊢ (𝐾 ∈ (SubGrp‘𝐸) → (0_g‘𝐸) = (0_g‘𝐿))
84	81, 83	syl 18	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (0_g‘𝐸) = (0_g‘𝐿))
85	77, 84	eqtrd 2800	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 ∪ 𝐵) × { 0 }) = (0_g‘𝐿))
86	85	adantr 486	. . . . . . . . . 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 2808	. . . . . . . . 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 426	. . . . . . . 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 245	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑎 ∈ 𝐾 → ((𝑎 ↾ 𝐵) = (𝐵 × { 0 }) → 𝑎 = (0_g‘𝐿))))
90	89	imp 412	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝑎 ↾ 𝐵) = (𝐵 × { 0 }) → 𝑎 = (0_g‘𝐿)))
91	57, 90	sylbid 243	. . . . 5 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝐹‘𝑎) = (0_g‘𝐷) → 𝑎 = (0_g‘𝐿)))
92	91	ralrimiva 3159	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑎 ∈ 𝐾 ((𝐹‘𝑎) = (0_g‘𝐷) → 𝑎 = (0_g‘𝐿)))
93		lmghm 21182	. . . . 5 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹 ∈ (𝐿 GrpHom 𝐷))
94	46, 12	ressbas2 17316	. . . . . . 7 ⊢ (𝐾 ⊆ 𝐺 → 𝐾 = (Base‘𝐿))
95	4, 94	ax-mp 5	. . . . . 6 ⊢ 𝐾 = (Base‘𝐿)
96		eqid 2765	. . . . . 6 ⊢ (0_g‘𝐿) = (0_g‘𝐿)
97		eqid 2765	. . . . . 6 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
98	95, 13, 96, 97	ghmf1 19340	. . . . 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 19	. . . 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 260	. . 3 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹:𝐾–1-1→(Base‘𝐷))
101		eqid 2765	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
102	101, 13	lmhmf 21185	. . . . 5 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:(Base‘𝐿)⟶(Base‘𝐷))
103		frn 6717	. . . . 5 ⊢ (𝐹:(Base‘𝐿)⟶(Base‘𝐷) → ran 𝐹 ⊆ (Base‘𝐷))
104	49, 102, 103	3syl 19	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ran 𝐹 ⊆ (Base‘𝐷))
105		reseq1 5974	. . . . . . 7 ⊢ (𝑥 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑥 ↾ 𝐵) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
106	11, 67, 13	pwselbasb 17559	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ 𝐵 ∈ V) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
107	17, 53, 106	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝑎 ∈ (Base‘𝐷) ↔ 𝑎:𝐵⟶(Base‘𝑅)))
108	107	biimpa 482	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎:𝐵⟶(Base‘𝑅))
109	26	fvexi 6899	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
110	109	fconst 6768	. . . . . . . . . . . . 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 486	. . . . . . . . . . . . . 14 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑅 ∈ Mnd)
113	67, 26	mndidcl 18829	. . . . . . . . . . . . . 14 ⊢ (𝑅 ∈ Mnd → 0 ∈ (Base‘𝑅))
114	112, 113	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 0 ∈ (Base‘𝑅))
115	114	snssd 4754	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → { 0 } ⊆ (Base‘𝑅))
116	111, 115	fssd 6727	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶(Base‘𝑅))
117		incom 4162	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
118		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
119	117, 118	eqtrid 2812	. . . . . . . . . . . 12 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐵 ∩ 𝐴) = ∅)
120	119	adantr 486	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵 ∩ 𝐴) = ∅)
121		fun 6744	. . . . . . . . . . 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 851	. . . . . . . . . 10 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐵 ∪ 𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)))
123		uncom 4112	. . . . . . . . . . 11 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
124		unidm 4111	. . . . . . . . . . 11 ⊢ ((Base‘𝑅) ∪ (Base‘𝑅)) = (Base‘𝑅)
125	123, 124	feq23i 6703	. . . . . . . . . 10 ⊢ ((𝑎 ∪ (𝐴 × { 0 })):(𝐵 ∪ 𝐴)⟶((Base‘𝑅) ∪ (Base‘𝑅)) ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅))
126	122, 125	sylib 221	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅))
127	10, 67, 12	pwselbasb 17559	. . . . . . . . . . 11 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅)))
128	127	3adant3 1150	. . . . . . . . . 10 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ↔ (𝑎 ∪ (𝐴 × { 0 })):(𝐴 ∪ 𝐵)⟶(Base‘𝑅)))
129	128	adantr 486	. . . . . . . . 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 260	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺)
131		simpl3 1212	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 ∩ 𝐵) = ∅)
132	117, 131	eqtrid 2812	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵 ∩ 𝐴) = ∅)
133		ffn 6709	. . . . . . . . . . . 12 ⊢ (𝑎:𝐵⟶(Base‘𝑅) → 𝑎 Fn 𝐵)
134		fnresdisj 6659	. . . . . . . . . . . 12 ⊢ (𝑎 Fn 𝐵 → ((𝐵 ∩ 𝐴) = ∅ ↔ (𝑎 ↾ 𝐴) = ∅))
135	108, 133, 134	3syl 19	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐵 ∩ 𝐴) = ∅ ↔ (𝑎 ↾ 𝐴) = ∅))
136	132, 135	mpbid 235	. . . . . . . . . 10 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ↾ 𝐴) = ∅)
137		fnconstg 6770	. . . . . . . . . . . 12 ⊢ ( 0 ∈ V → (𝐴 × { 0 }) Fn 𝐴)
138		fnresdm 6658	. . . . . . . . . . . 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 4123	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴)) = (∅ ∪ (𝐴 × { 0 })))
142		resundir 5995	. . . . . . . . 9 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = ((𝑎 ↾ 𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴))
143		uncom 4112	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐴 × { 0 })) = ((𝐴 × { 0 }) ∪ ∅)
144		un0 4351	. . . . . . . . . 10 ⊢ ((𝐴 × { 0 }) ∪ ∅) = (𝐴 × { 0 })
145	143, 144	eqtr2i 2789	. . . . . . . . 9 ⊢ (𝐴 × { 0 }) = (∅ ∪ (𝐴 × { 0 }))
146	141, 142, 145	3eqtr4g 2825	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 }))
147		reseq1 5974	. . . . . . . . . 10 ⊢ (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑦 ↾ 𝐴) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴))
148	147	eqeq1d 2767	. . . . . . . . 9 ⊢ (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
149	148, 2	elrab2 3656	. . . . . . . 8 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾 ↔ ((𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐺 ∧ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 })))
150	130, 146, 149	sylanbrc 595	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾)
151	130	resexd 6029	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
152	1, 105, 150, 151	fvmptd3 7017	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
153		resundir 5995	. . . . . . 7 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵))
154		fnresdm 6658	. . . . . . . . . 10 ⊢ (𝑎 Fn 𝐵 → (𝑎 ↾ 𝐵) = 𝑎)
155	108, 133, 154	3syl 19	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ↾ 𝐵) = 𝑎)
156		ffn 6709	. . . . . . . . . . . . 13 ⊢ ((𝐴 × { 0 }):𝐴⟶{ 0 } → (𝐴 × { 0 }) Fn 𝐴)
157		fnresdisj 6659	. . . . . . . . . . . . 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 219	. . . . . . . . . . 11 ⊢ ((𝐴 ∩ 𝐵) = ∅ → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
160	159	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
161	160	adantr 486	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝐴 × { 0 }) ↾ 𝐵) = ∅)
162	155, 161	uneq12d 4123	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = (𝑎 ∪ ∅))
163		un0 4351	. . . . . . . 8 ⊢ (𝑎 ∪ ∅) = 𝑎
164	162, 163	eqtrdi 2816	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = 𝑎)
165	153, 164	eqtrid 2812	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = 𝑎)
166	152, 165	eqtrd 2800	. . . . 5 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = 𝑎)
167	95, 13	lmhmf 21185	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:𝐾⟶(Base‘𝐷))
168		ffn 6709	. . . . . . . 8 ⊢ (𝐹:𝐾⟶(Base‘𝐷) → 𝐹 Fn 𝐾)
169	49, 167, 168	3syl 19	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 Fn 𝐾)
170	169	adantr 486	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝐹 Fn 𝐾)
171		fnfvelrn 7079	. . . . . 6 ⊢ ((𝐹 Fn 𝐾 ∧ (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
172	170, 150, 171	syl2anc 596	. . . . 5 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) ∈ ran 𝐹)
173	166, 172	eqeltrrd 2866	. . . 4 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎 ∈ ran 𝐹)
174	104, 173	eqelssd 3959	. . 3 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ran 𝐹 = (Base‘𝐷))
175		dff1o5 6834	. . 3 ⊢ (𝐹:𝐾–1-1-onto→(Base‘𝐷) ↔ (𝐹:𝐾–1-1→(Base‘𝐷) ∧ ran 𝐹 = (Base‘𝐷)))
176	100, 174, 175	sylanbrc 595	. 2 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹:𝐾–1-1-onto→(Base‘𝐷))
177	95, 13	islmim 21213	. 2 ⊢ (𝐹 ∈ (𝐿 LMIso 𝐷) ↔ (𝐹 ∈ (𝐿 LMHom 𝐷) ∧ 𝐹:𝐾–1-1-onto→(Base‘𝐷)))
178	49, 176, 177	sylanbrc 595	1 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 ∈ (𝐿 LMIso 𝐷))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3081 {crab 3418 Vcvv 3457 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4286 {csn 4591 ↦ cmpt 5194 × cxp 5661 ^◡ccnv 5662 ran crn 5664 ↾ cres 5665 “ cima 5666 Fn wfn 6535 ⟶wf 6536 –1-1→wf1 6537 –1-1-onto→wf1o 6539 ‘cfv 6540 (class class class)co 7416 Basecbs 17287 ↾_s cress 17308 0_gc0g 17510 ↑_s cpws 17517 Mndcmnd 18814 Grpcgrp 19024 SubGrpcsubg 19210 GrpHom cghm 19307 LModclmod 21011 LSubSpclss 21082 LMHom clmhm 21170 LMIso clmim 21171

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7738 ax-cnex 11167 ax-resscn 11168 ax-1cn 11169 ax-icn 11170 ax-addcl 11171 ax-addrcl 11172 ax-mulcl 11173 ax-mulrcl 11174 ax-mulcom 11175 ax-addass 11176 ax-mulass 11177 ax-distr 11178 ax-i2m1 11179 ax-1ne0 11180 ax-1rid 11181 ax-rnegex 11182 ax-rrecex 11183 ax-cnre 11184 ax-pre-lttri 11185 ax-pre-lttrn 11186 ax-pre-ltadd 11187 ax-pre-mulgt0 11188

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-sup 9405 df-pnf 11256 df-mnf 11257 df-xr 11258 df-ltxr 11259 df-le 11260 df-sub 11454 df-neg 11455 df-nn 12245 df-2 12314 df-3 12315 df-4 12316 df-5 12317 df-6 12318 df-7 12319 df-8 12320 df-9 12321 df-n0 12516 df-z 12603 df-dec 12724 df-uz 12875 df-fz 13548 df-struct 17225 df-sets 17242 df-slot 17260 df-ndx 17272 df-base 17288 df-ress 17309 df-plusg 17341 df-mulr 17342 df-sca 17344 df-vsca 17345 df-ip 17346 df-tset 17347 df-ple 17348 df-ds 17350 df-hom 17352 df-cco 17353 df-0g 17512 df-prds 17518 df-pws 17520 df-mgm 18716 df-sgrp 18799 df-mnd 18815 df-grp 19027 df-minusg 19028 df-sbg 19029 df-subg 19213 df-ghm 19308 df-cmn 19876 df-abl 19877 df-mgp 20241 df-rng 20255 df-ur 20288 df-ring 20341 df-lmod 21013 df-lss 21083 df-lmhm 21173 df-lmim 21174

This theorem is used by: pwslnmlem2 43853

W3C validator