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Theorem mrcmndind 17366

Description: (( From SO's determinants branch )). TODO: Appropriate description to be added! (Contributed by SO, 14-Jul-2018.)

**Hypotheses**
Ref	Expression
mrcmndind.ch	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
mrcmndind.th	⊢ (𝑥 = (𝑦 + 𝑧) → (𝜓 ↔ 𝜃))
mrcmndind.ta	⊢ (𝑥 = 0 → (𝜓 ↔ 𝜏))
mrcmndind.et	⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))
mrcmndind.0g	⊢ 0 = (0_g‘𝑀)
mrcmndind.pg	⊢ + = (+_g‘𝑀)
mrcmndind.b	⊢ 𝐵 = (Base‘𝑀)
mrcmndind.m	⊢ (𝜑 → 𝑀 ∈ Mnd)
mrcmndind.g	⊢ (𝜑 → 𝐺 ⊆ 𝐵)
mrcmndind.k	⊢ (𝜑 → 𝐵 = ((mrCls‘(SubMnd‘𝑀))‘𝐺))
mrcmndind.i1	⊢ (𝜑 → 𝜏)
mrcmndind.i2	⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐺) ∧ 𝜒) → 𝜃)
mrcmndind.a	⊢ (𝜑 → 𝐴 ∈ 𝐵)

**Assertion**
Ref	Expression
mrcmndind	⊢ (𝜑 → 𝜂)

Distinct variable groups: 𝜑,𝑥,𝑦,𝑧 𝜓,𝑦,𝑧 𝜒,𝑥,𝑧 𝜃,𝑥 𝑥, 0 𝑥,𝐴 𝜏,𝑥 𝜂,𝑥 𝑦,𝐺,𝑧 𝑦,𝐵,𝑧 𝑥, + ,𝑦,𝑧 Allowed substitution hints: 𝜓(𝑥) 𝜒(𝑦) 𝜃(𝑦,𝑧) 𝜏(𝑦,𝑧) 𝜂(𝑦,𝑧) 𝐴(𝑦,𝑧) 𝐵(𝑥) 𝐺(𝑥) 𝑀(𝑥,𝑦,𝑧) 0 (𝑦,𝑧)

Proof of Theorem mrcmndind Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrcmndind.i1	. . . 4 ⊢ (𝜑 → 𝜏)
2		mrcmndind.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Mnd)
3		mrcmndind.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
4		mrcmndind.0g	. . . . . . 7 ⊢ 0 = (0_g‘𝑀)
5	3, 4	mndidcl 17308	. . . . . 6 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵)
6	2, 5	syl 17	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
7		mrcmndind.ta	. . . . . 6 ⊢ (𝑥 = 0 → (𝜓 ↔ 𝜏))
8	7	sbcieg 3468	. . . . 5 ⊢ ( 0 ∈ 𝐵 → ([ 0 / 𝑥]𝜓 ↔ 𝜏))
9	6, 8	syl 17	. . . 4 ⊢ (𝜑 → ([ 0 / 𝑥]𝜓 ↔ 𝜏))
10	1, 9	mpbird 247	. . 3 ⊢ (𝜑 → [ 0 / 𝑥]𝜓)
11		mrcmndind.k	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((mrCls‘(SubMnd‘𝑀))‘𝐺))
12	3	submacs 17365	. . . . . . . . . 10 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
13	2, 12	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
14	13	acsmred 16317	. . . . . . . 8 ⊢ (𝜑 → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
15		mrcmndind.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ 𝐵)
16		eleq1 2689	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ 𝐵 ↔ 𝑎 ∈ 𝐵))
17	16	anbi2d 740	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑎 → (((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑎 ∈ 𝐵)))
18		vex 3203	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
19		mrcmndind.ch	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
20	18, 19	sbcie 3470	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜒)
21		dfsbcq 3437	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → ([𝑦 / 𝑥]𝜓 ↔ [𝑎 / 𝑥]𝜓))
22	20, 21	syl5bbr 274	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → (𝜒 ↔ [𝑎 / 𝑥]𝜓))
23		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → (𝑦 + 𝑏) = (𝑎 + 𝑏))
24	23	sbceq1d 3440	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → ([(𝑦 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + 𝑏) / 𝑥]𝜓))
25	22, 24	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑎 → ((𝜒 → [(𝑦 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)))
26	17, 25	imbi12d 334	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑎 → ((((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) → (𝜒 → [(𝑦 + 𝑏) / 𝑥]𝜓)) ↔ (((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑎 ∈ 𝐵) → ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓))))
27		eleq1 2689	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑏 → (𝑧 ∈ 𝐺 ↔ 𝑏 ∈ 𝐺))
28	27	anbi2d 740	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑏 → ((𝜑 ∧ 𝑧 ∈ 𝐺) ↔ (𝜑 ∧ 𝑏 ∈ 𝐺)))
29	28	anbi1d 741	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑏 → (((𝜑 ∧ 𝑧 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵)))
30		ovex 6678	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 + 𝑧) ∈ V
31		mrcmndind.th	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝜓 ↔ 𝜃))
32	30, 31	sbcie 3470	. . . . . . . . . . . . . . 15 ⊢ ([(𝑦 + 𝑧) / 𝑥]𝜓 ↔ 𝜃)
33		oveq2 6658	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑏 → (𝑦 + 𝑧) = (𝑦 + 𝑏))
34	33	sbceq1d 3440	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑏 → ([(𝑦 + 𝑧) / 𝑥]𝜓 ↔ [(𝑦 + 𝑏) / 𝑥]𝜓))
35	32, 34	syl5bbr 274	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑏 → (𝜃 ↔ [(𝑦 + 𝑏) / 𝑥]𝜓))
36	35	imbi2d 330	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑏 → ((𝜒 → 𝜃) ↔ (𝜒 → [(𝑦 + 𝑏) / 𝑥]𝜓)))
37	29, 36	imbi12d 334	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑏 → ((((𝜑 ∧ 𝑧 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) → (𝜒 → 𝜃)) ↔ (((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) → (𝜒 → [(𝑦 + 𝑏) / 𝑥]𝜓))))
38		mrcmndind.i2	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐺) ∧ 𝜒) → 𝜃)
39	38	ex 450	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐺) → (𝜒 → 𝜃))
40	39	3expa 1265	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐺) → (𝜒 → 𝜃))
41	40	an32s 846	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) → (𝜒 → 𝜃))
42	37, 41	chvarv 2263	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) → (𝜒 → [(𝑦 + 𝑏) / 𝑥]𝜓))
43	26, 42	chvarv 2263	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑎 ∈ 𝐵) → ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓))
44	43	ralrimiva 2966	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐺) → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓))
45	15, 44	ssrabdv 3681	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ⊆ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)})
46		mrcmndind.pg	. . . . . . . . 9 ⊢ + = (+_g‘𝑀)
47	3, 46, 4	mndrid 17312	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ Mnd ∧ 𝑎 ∈ 𝐵) → (𝑎 + 0 ) = 𝑎)
48	2, 47	sylan 488	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 + 0 ) = 𝑎)
49	48	sbceq1d 3440	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ([(𝑎 + 0 ) / 𝑥]𝜓 ↔ [𝑎 / 𝑥]𝜓))
50	49	biimprd 238	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ([𝑎 / 𝑥]𝜓 → [(𝑎 + 0 ) / 𝑥]𝜓))
51	50	ralrimiva 2966	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 0 ) / 𝑥]𝜓))
52		simprrl 804	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵) ∧ (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑐) / 𝑥]𝜓) ∧ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓)))) → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑐) / 𝑥]𝜓))
53	2	ad2antrr 762	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑀 ∈ Mnd)
54		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
55		simplrl 800	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑐 ∈ 𝐵)
56	3, 46	mndcl 17301	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ Mnd ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) → (𝑏 + 𝑐) ∈ 𝐵)
57	53, 54, 55, 56	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝑏 + 𝑐) ∈ 𝐵)
58		simpr 477	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) ∧ 𝑎 = (𝑏 + 𝑐)) → 𝑎 = (𝑏 + 𝑐))
59	58	sbceq1d 3440	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) ∧ 𝑎 = (𝑏 + 𝑐)) → ([𝑎 / 𝑥]𝜓 ↔ [(𝑏 + 𝑐) / 𝑥]𝜓))
60		oveq1 6657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝑏 + 𝑐) → (𝑎 + 𝑑) = ((𝑏 + 𝑐) + 𝑑))
61		simplrr 801	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑑 ∈ 𝐵)
62	3, 46	mndass 17302	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ Mnd ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) → ((𝑏 + 𝑐) + 𝑑) = (𝑏 + (𝑐 + 𝑑)))
63	53, 54, 55, 61, 62	syl13anc 1328	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑏 + 𝑐) + 𝑑) = (𝑏 + (𝑐 + 𝑑)))
64	60, 63	sylan9eqr 2678	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) ∧ 𝑎 = (𝑏 + 𝑐)) → (𝑎 + 𝑑) = (𝑏 + (𝑐 + 𝑑)))
65	64	sbceq1d 3440	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) ∧ 𝑎 = (𝑏 + 𝑐)) → ([(𝑎 + 𝑑) / 𝑥]𝜓 ↔ [(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓))
66	59, 65	imbi12d 334	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) ∧ 𝑎 = (𝑏 + 𝑐)) → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓) ↔ ([(𝑏 + 𝑐) / 𝑥]𝜓 → [(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
67	57, 66	rspcdv 3312	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓) → ([(𝑏 + 𝑐) / 𝑥]𝜓 → [(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
68	67	ralrimdva 2969	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓) → ∀𝑏 ∈ 𝐵 ([(𝑏 + 𝑐) / 𝑥]𝜓 → [(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
69	68	impr 649	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓))) → ∀𝑏 ∈ 𝐵 ([(𝑏 + 𝑐) / 𝑥]𝜓 → [(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓))
70		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑎 → (𝑏 + 𝑐) = (𝑎 + 𝑐))
71	70	sbceq1d 3440	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑎 → ([(𝑏 + 𝑐) / 𝑥]𝜓 ↔ [(𝑎 + 𝑐) / 𝑥]𝜓))
72		oveq1 6657	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑎 → (𝑏 + (𝑐 + 𝑑)) = (𝑎 + (𝑐 + 𝑑)))
73	72	sbceq1d 3440	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑎 → ([(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓 ↔ [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓))
74	71, 73	imbi12d 334	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑎 → (([(𝑏 + 𝑐) / 𝑥]𝜓 → [(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓) ↔ ([(𝑎 + 𝑐) / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
75	74	cbvralv 3171	. . . . . . . . . . . 12 ⊢ (∀𝑏 ∈ 𝐵 ([(𝑏 + 𝑐) / 𝑥]𝜓 → [(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([(𝑎 + 𝑐) / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓))
76	69, 75	sylib 208	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵) ∧ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓))) → ∀𝑎 ∈ 𝐵 ([(𝑎 + 𝑐) / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓))
77	76	adantrrl 760	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵) ∧ (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑐) / 𝑥]𝜓) ∧ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓)))) → ∀𝑎 ∈ 𝐵 ([(𝑎 + 𝑐) / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓))
78		imim1 83	. . . . . . . . . . 11 ⊢ (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑐) / 𝑥]𝜓) → (([(𝑎 + 𝑐) / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓) → ([𝑎 / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
79	78	ral2imi 2947	. . . . . . . . . 10 ⊢ (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑐) / 𝑥]𝜓) → (∀𝑎 ∈ 𝐵 ([(𝑎 + 𝑐) / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓) → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
80	52, 77, 79	sylc 65	. . . . . . . . 9 ⊢ ((𝜑 ∧ ((𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵) ∧ (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑐) / 𝑥]𝜓) ∧ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓)))) → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓))
81		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑏 = 0 → (𝑎 + 𝑏) = (𝑎 + 0 ))
82	81	sbceq1d 3440	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → ([(𝑎 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + 0 ) / 𝑥]𝜓))
83	82	imbi2d 330	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + 0 ) / 𝑥]𝜓)))
84	83	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑏 = 0 → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 0 ) / 𝑥]𝜓)))
85		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑐 → (𝑎 + 𝑏) = (𝑎 + 𝑐))
86	85	sbceq1d 3440	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ([(𝑎 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + 𝑐) / 𝑥]𝜓))
87	86	imbi2d 330	. . . . . . . . . 10 ⊢ (𝑏 = 𝑐 → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑐) / 𝑥]𝜓)))
88	87	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑐) / 𝑥]𝜓)))
89		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → (𝑎 + 𝑏) = (𝑎 + 𝑑))
90	89	sbceq1d 3440	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → ([(𝑎 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + 𝑑) / 𝑥]𝜓))
91	90	imbi2d 330	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓)))
92	91	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓)))
93		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑐 + 𝑑) → (𝑎 + 𝑏) = (𝑎 + (𝑐 + 𝑑)))
94	93	sbceq1d 3440	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑐 + 𝑑) → ([(𝑎 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓))
95	94	imbi2d 330	. . . . . . . . . 10 ⊢ (𝑏 = (𝑐 + 𝑑) → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
96	95	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑏 = (𝑐 + 𝑑) → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
97	3, 46, 4, 2, 51, 80, 84, 88, 92, 96	issubmd 17349	. . . . . . . 8 ⊢ (𝜑 → {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)} ∈ (SubMnd‘𝑀))
98		eqid 2622	. . . . . . . . 9 ⊢ (mrCls‘(SubMnd‘𝑀)) = (mrCls‘(SubMnd‘𝑀))
99	98	mrcsscl 16280	. . . . . . . 8 ⊢ (((SubMnd‘𝑀) ∈ (Moore‘𝐵) ∧ 𝐺 ⊆ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)} ∧ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)} ∈ (SubMnd‘𝑀)) → ((mrCls‘(SubMnd‘𝑀))‘𝐺) ⊆ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)})
100	14, 45, 97, 99	syl3anc 1326	. . . . . . 7 ⊢ (𝜑 → ((mrCls‘(SubMnd‘𝑀))‘𝐺) ⊆ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)})
101	11, 100	eqsstrd 3639	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)})
102		mrcmndind.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
103	101, 102	sseldd 3604	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)})
104		oveq2 6658	. . . . . . . . . 10 ⊢ (𝑏 = 𝐴 → (𝑎 + 𝑏) = (𝑎 + 𝐴))
105	104	sbceq1d 3440	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → ([(𝑎 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + 𝐴) / 𝑥]𝜓))
106	105	imbi2d 330	. . . . . . . 8 ⊢ (𝑏 = 𝐴 → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓)))
107	106	ralbidv 2986	. . . . . . 7 ⊢ (𝑏 = 𝐴 → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓)))
108	107	elrab 3363	. . . . . 6 ⊢ (𝐴 ∈ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)} ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓)))
109	108	simprbi 480	. . . . 5 ⊢ (𝐴 ∈ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)} → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓))
110	103, 109	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓))
111		dfsbcq 3437	. . . . . 6 ⊢ (𝑎 = 0 → ([𝑎 / 𝑥]𝜓 ↔ [ 0 / 𝑥]𝜓))
112		oveq1 6657	. . . . . . 7 ⊢ (𝑎 = 0 → (𝑎 + 𝐴) = ( 0 + 𝐴))
113	112	sbceq1d 3440	. . . . . 6 ⊢ (𝑎 = 0 → ([(𝑎 + 𝐴) / 𝑥]𝜓 ↔ [( 0 + 𝐴) / 𝑥]𝜓))
114	111, 113	imbi12d 334	. . . . 5 ⊢ (𝑎 = 0 → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓) ↔ ([ 0 / 𝑥]𝜓 → [( 0 + 𝐴) / 𝑥]𝜓)))
115	114	rspcva 3307	. . . 4 ⊢ (( 0 ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓)) → ([ 0 / 𝑥]𝜓 → [( 0 + 𝐴) / 𝑥]𝜓))
116	6, 110, 115	syl2anc 693	. . 3 ⊢ (𝜑 → ([ 0 / 𝑥]𝜓 → [( 0 + 𝐴) / 𝑥]𝜓))
117	10, 116	mpd 15	. 2 ⊢ (𝜑 → [( 0 + 𝐴) / 𝑥]𝜓)
118	3, 46, 4	mndlid 17311	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → ( 0 + 𝐴) = 𝐴)
119	2, 102, 118	syl2anc 693	. . . 4 ⊢ (𝜑 → ( 0 + 𝐴) = 𝐴)
120	119	sbceq1d 3440	. . 3 ⊢ (𝜑 → ([( 0 + 𝐴) / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓))
121		mrcmndind.et	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))
122	121	sbcieg 3468	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜂))
123	102, 122	syl 17	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜂))
124	120, 123	bitrd 268	. 2 ⊢ (𝜑 → ([( 0 + 𝐴) / 𝑥]𝜓 ↔ 𝜂))
125	117, 124	mpbid 222	1 ⊢ (𝜑 → 𝜂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 [wsbc 3435 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +_gcplusg 15941 0_gc0g 16100 Moorecmre 16242 mrClscmrc 16243 ACScacs 16245 Mndcmnd 17294 SubMndcsubmnd 17334

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-fin 7959 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336

This theorem is referenced by: mdetunilem7 20424

W3C validator