pwssplit4 - Mathbox for Stefan O'Rear

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Theorem pwssplit4 43930

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 4028	. . . . . 6 ⊢ {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (𝐴 × { 0 })} ⊆ 𝐺
4	2, 3	eqsstri 3977	. . . . 5 ⊢ 𝐾 ⊆ 𝐺
5		resmpt 6033	. . . . 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 2786	. . 3 ⊢ 𝐹 = ((𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) ↾ 𝐾)
8		ssun2 4125	. . . . . 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 2760	. . . . . 6 ⊢ (Base‘𝐷) = (Base‘𝐷)
14		eqid 2760	. . . . . 6 ⊢ (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵)) = (𝑥 ∈ 𝐺 ↦ (𝑥 ↾ 𝐵))
15	10, 11, 12, 13, 14	pwssplit3 21245	. . . . 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 21051	. . . . . . . . . 10 ⊢ (𝑅 ∈ LMod → 𝑅 ∈ Grp)
19		grpmnd 19064	. . . . . . . . . 10 ⊢ (𝑅 ∈ Grp → 𝑅 ∈ Mnd)
20	17, 18, 19	3syl 19	. . . . . . . . 9 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝑅 ∈ Mnd)
21		ssun1 4124	. . . . . . . . . . 11 ⊢ 𝐴 ⊆ (𝐴 ∪ 𝐵)
22		ssexg 5284	. . . . . . . . . . 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 18880	. . . . . . . . 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 2771	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ((𝑦 ↾ 𝐴) = (𝐴 × { 0 }) ↔ (𝑦 ↾ 𝐴) = (0_g‘𝐶)))
30	29	rabbidv 3419	. . . . . 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 2807	. . . . 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 2760	. . . . . . . 8 ⊢ (Base‘𝐶) = (Base‘𝐶)
34		eqid 2760	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) = (𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴))
35	10, 25, 12, 33, 34	pwssplit3 21245	. . . . . . 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 6891	. . . . . . . . 9 ⊢ (0_g‘𝐶) ∈ V
38	34	mptiniseg 6235	. . . . . . . . 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 2769	. . . . . . 7 ⊢ {𝑦 ∈ 𝐺 ∣ (𝑦 ↾ 𝐴) = (0_g‘𝐶)} = (^◡(𝑦 ∈ 𝐺 ↦ (𝑦 ↾ 𝐴)) “ {(0_g‘𝐶)})
41		eqid 2760	. . . . . . 7 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
42		eqid 2760	. . . . . . 7 ⊢ (LSubSp‘𝐸) = (LSubSp‘𝐸)
43	40, 41, 42	lmhmkerlss 21235	. . . . . 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 2860	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐾 ∈ (LSubSp‘𝐸))
46		pwssplit4.l	. . . . 5 ⊢ 𝐿 = (𝐸 ↾_s 𝐾)
47	42, 46	reslmhm 21236	. . . 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 2864	. 2 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → 𝐹 ∈ (𝐿 LMHom 𝐷))
50	1	fvtresfn 6989	. . . . . . 7 ⊢ (𝑎 ∈ 𝐾 → (𝐹‘𝑎) = (𝑎 ↾ 𝐵))
51		ssexg 5284	. . . . . . . . . . 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 18880	. . . . . . . . 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 2766	. . . . . . 7 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (0_g‘𝐷) = (𝐵 × { 0 }))
57	50, 56	eqeqan12rd 2775	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ 𝐾) → ((𝐹‘𝑎) = (0_g‘𝐷) ↔ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })))
58		reseq1 5966	. . . . . . . . . 10 ⊢ (𝑦 = 𝑎 → (𝑦 ↾ 𝐴) = (𝑎 ↾ 𝐴))
59	58	eqeq1d 2762	. . . . . . . . 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 4110	. . . . . . . . . . . . 13 ⊢ (((𝑎 ↾ 𝐴) = (𝐴 × { 0 }) ∧ (𝑎 ↾ 𝐵) = (𝐵 × { 0 })) → ((𝑎 ↾ 𝐴) ∪ (𝑎 ↾ 𝐵)) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 })))
62		resundi 5986	. . . . . . . . . . . . 13 ⊢ (𝑎 ↾ (𝐴 ∪ 𝐵)) = ((𝑎 ↾ 𝐴) ∪ (𝑎 ↾ 𝐵))
63		xpundir 5725	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∪ 𝐵) × { 0 }) = ((𝐴 × { 0 }) ∪ (𝐵 × { 0 }))
64	61, 62, 63	3eqtr4g 2820	. . . . . . . . . . . 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 2760	. . . . . . . . . . . 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 17574	. . . . . . . . . . 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 6702	. . . . . . . . . . 11 ⊢ (𝑎:(𝐴 ∪ 𝐵)⟶(Base‘𝑅) → 𝑎 Fn (𝐴 ∪ 𝐵))
74		fnresdm 6651	. . . . . . . . . . 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 18880	. . . . . . . . . . . . 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 21154	. . . . . . . . . . . . . . 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 21141	. . . . . . . . . . . . . 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 2760	. . . . . . . . . . . . . 14 ⊢ (0_g‘𝐸) = (0_g‘𝐸)
83	46, 82	subg0 19255	. . . . . . . . . . . . 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 2795	. . . . . . . . . . 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 2803	. . . . . . . . 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 3154	. . . 4 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ∀𝑎 ∈ 𝐾 ((𝐹‘𝑎) = (0_g‘𝐷) → 𝑎 = (0_g‘𝐿)))
93		lmghm 21215	. . . . 5 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹 ∈ (𝐿 GrpHom 𝐷))
94	46, 12	ressbas2 17330	. . . . . . 7 ⊢ (𝐾 ⊆ 𝐺 → 𝐾 = (Base‘𝐿))
95	4, 94	ax-mp 5	. . . . . 6 ⊢ 𝐾 = (Base‘𝐿)
96		eqid 2760	. . . . . 6 ⊢ (0_g‘𝐿) = (0_g‘𝐿)
97		eqid 2760	. . . . . 6 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
98	95, 13, 96, 97	ghmf1 19373	. . . . 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 2760	. . . . . 6 ⊢ (Base‘𝐿) = (Base‘𝐿)
102	101, 13	lmhmf 21218	. . . . 5 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:(Base‘𝐿)⟶(Base‘𝐷))
103		frn 6710	. . . . 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 5966	. . . . . . 7 ⊢ (𝑥 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑥 ↾ 𝐵) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
106	11, 67, 13	pwselbasb 17573	. . . . . . . . . . . . 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 6892	. . . . . . . . . . . . . 14 ⊢ 0 ∈ V
110	109	fconst 6761	. . . . . . . . . . . . 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 18852	. . . . . . . . . . . . . 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 4747	. . . . . . . . . . . 12 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → { 0 } ⊆ (Base‘𝑅))
116	111, 115	fssd 6720	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐴 × { 0 }):𝐴⟶(Base‘𝑅))
117		incom 4155	. . . . . . . . . . . . 13 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
118		simp3 1156	. . . . . . . . . . . . 13 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → (𝐴 ∩ 𝐵) = ∅)
119	117, 118	eqtrid 2807	. . . . . . . . . . . 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 6737	. . . . . . . . . . 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 4105	. . . . . . . . . . 11 ⊢ (𝐵 ∪ 𝐴) = (𝐴 ∪ 𝐵)
124		unidm 4104	. . . . . . . . . . 11 ⊢ ((Base‘𝑅) ∪ (Base‘𝑅)) = (Base‘𝑅)
125	123, 124	feq23i 6696	. . . . . . . . . 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 17573	. . . . . . . . . . 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 2807	. . . . . . . . . . 11 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐵 ∩ 𝐴) = ∅)
133		ffn 6702	. . . . . . . . . . . 12 ⊢ (𝑎:𝐵⟶(Base‘𝑅) → 𝑎 Fn 𝐵)
134		fnresdisj 6652	. . . . . . . . . . . 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 6763	. . . . . . . . . . . 12 ⊢ ( 0 ∈ V → (𝐴 × { 0 }) Fn 𝐴)
138		fnresdm 6651	. . . . . . . . . . . 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 4116	. . . . . . . . 9 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴)) = (∅ ∪ (𝐴 × { 0 })))
142		resundir 5987	. . . . . . . . 9 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = ((𝑎 ↾ 𝐴) ∪ ((𝐴 × { 0 }) ↾ 𝐴))
143		uncom 4105	. . . . . . . . . 10 ⊢ (∅ ∪ (𝐴 × { 0 })) = ((𝐴 × { 0 }) ∪ ∅)
144		un0 4344	. . . . . . . . . 10 ⊢ ((𝐴 × { 0 }) ∪ ∅) = (𝐴 × { 0 })
145	143, 144	eqtr2i 2784	. . . . . . . . 9 ⊢ (𝐴 × { 0 }) = (∅ ∪ (𝐴 × { 0 }))
146	141, 142, 145	3eqtr4g 2820	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴) = (𝐴 × { 0 }))
147		reseq1 5966	. . . . . . . . . 10 ⊢ (𝑦 = (𝑎 ∪ (𝐴 × { 0 })) → (𝑦 ↾ 𝐴) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐴))
148	147	eqeq1d 2762	. . . . . . . . 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 595	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝑎 ∪ (𝐴 × { 0 })) ∈ 𝐾)
151	130	resexd 6021	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) ∈ V)
152	1, 105, 150, 151	fvmptd3 7010	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵))
153		resundir 5987	. . . . . . 7 ⊢ ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵))
154		fnresdm 6651	. . . . . . . . . 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 6702	. . . . . . . . . . . . 13 ⊢ ((𝐴 × { 0 }):𝐴⟶{ 0 } → (𝐴 × { 0 }) Fn 𝐴)
157		fnresdisj 6652	. . . . . . . . . . . . 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 4116	. . . . . . . 8 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = (𝑎 ∪ ∅))
163		un0 4344	. . . . . . . 8 ⊢ (𝑎 ∪ ∅) = 𝑎
164	162, 163	eqtrdi 2811	. . . . . . 7 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ↾ 𝐵) ∪ ((𝐴 × { 0 }) ↾ 𝐵)) = 𝑎)
165	153, 164	eqtrid 2807	. . . . . 6 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → ((𝑎 ∪ (𝐴 × { 0 })) ↾ 𝐵) = 𝑎)
166	152, 165	eqtrd 2795	. . . . 5 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → (𝐹‘(𝑎 ∪ (𝐴 × { 0 }))) = 𝑎)
167	95, 13	lmhmf 21218	. . . . . . . 8 ⊢ (𝐹 ∈ (𝐿 LMHom 𝐷) → 𝐹:𝐾⟶(Base‘𝐷))
168		ffn 6702	. . . . . . . 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 7073	. . . . . 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 2861	. . . 4 ⊢ (((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) ∧ 𝑎 ∈ (Base‘𝐷)) → 𝑎 ∈ ran 𝐹)
174	104, 173	eqelssd 3952	. . 3 ⊢ ((𝑅 ∈ LMod ∧ (𝐴 ∪ 𝐵) ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) = ∅) → ran 𝐹 = (Base‘𝐷))
175		dff1o5 6827	. . 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 21246	. 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 2145 ∀wral 3076 {crab 3412 Vcvv 3450 ∪ cun 3897 ∩ cin 3898 ⊆ wss 3899 ∅c0 4279 {csn 4584 ↦ cmpt 5186 × cxp 5653 ^◡ccnv 5654 ran crn 5656 ↾ cres 5657 “ cima 5658 Fn wfn 6528 ⟶wf 6529 –1-1→wf1 6530 –1-1-onto→wf1o 6532 ‘cfv 6533 (class class class)co 7413 Basecbs 17301 ↾_s cress 17322 0_gc0g 17524 ↑_s cpws 17531 Mndcmnd 18836 Grpcgrp 19057 SubGrpcsubg 19243 GrpHom cghm 19340 LModclmod 21044 LSubSpclss 21115 LMHom clmhm 21203 LMIso clmim 21204

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201

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 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-ixp 8905 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-sup 9412 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12529 df-z 12616 df-dec 12737 df-uz 12888 df-fz 13562 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-sca 17358 df-vsca 17359 df-ip 17360 df-tset 17361 df-ple 17362 df-ds 17364 df-hom 17366 df-cco 17367 df-0g 17526 df-prds 17532 df-pws 17534 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-grp 19060 df-minusg 19061 df-sbg 19062 df-subg 19246 df-ghm 19341 df-cmn 19909 df-abl 19910 df-mgp 20274 df-rng 20288 df-ur 20321 df-ring 20374 df-lmod 21046 df-lss 21116 df-lmhm 21206 df-lmim 21207

This theorem is used by: pwslnmlem2 43934

W3C validator