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Theorem mndind 18709

Description: Induction in a monoid. In this theorem, 𝜓(𝑥) is the "generic" proposition to be be proved (the first four hypotheses tell its values at y, y+z, 0, A respectively). The two induction hypotheses mndind.i1 and mndind.i2 tell that it is true at 0, that if it is true at y then it is true at y+z (provided z is in 𝐺). The hypothesis mndind.k tells that 𝐺 is generating. (Contributed by SO, 14-Jul-2018.)

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

**Assertion**
Ref	Expression
mndind	⊢ (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 mndind Dummy variables 𝑎 𝑏 𝑐 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mndind.i1	. . . 4 ⊢ (𝜑 → 𝜏)
2		mndind.m	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ Mnd)
3		mndind.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
4		mndind.0g	. . . . . . 7 ⊢ 0 = (0_g‘𝑀)
5	3, 4	mndidcl 18640	. . . . . 6 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵)
6	2, 5	syl 17	. . . . 5 ⊢ (𝜑 → 0 ∈ 𝐵)
7		mndind.ta	. . . . . 6 ⊢ (𝑥 = 0 → (𝜓 ↔ 𝜏))
8	7	sbcieg 3818	. . . . 5 ⊢ ( 0 ∈ 𝐵 → ([ 0 / 𝑥]𝜓 ↔ 𝜏))
9	6, 8	syl 17	. . . 4 ⊢ (𝜑 → ([ 0 / 𝑥]𝜓 ↔ 𝜏))
10	1, 9	mpbird 257	. . 3 ⊢ (𝜑 → [ 0 / 𝑥]𝜓)
11		dfsbcq 3780	. . . . 5 ⊢ (𝑎 = 0 → ([𝑎 / 𝑥]𝜓 ↔ [ 0 / 𝑥]𝜓))
12		oveq1 7416	. . . . . 6 ⊢ (𝑎 = 0 → (𝑎 + 𝐴) = ( 0 + 𝐴))
13	12	sbceq1d 3783	. . . . 5 ⊢ (𝑎 = 0 → ([(𝑎 + 𝐴) / 𝑥]𝜓 ↔ [( 0 + 𝐴) / 𝑥]𝜓))
14	11, 13	imbi12d 345	. . . 4 ⊢ (𝑎 = 0 → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓) ↔ ([ 0 / 𝑥]𝜓 → [( 0 + 𝐴) / 𝑥]𝜓)))
15		mndind.k	. . . . . . 7 ⊢ (𝜑 → 𝐵 = ((mrCls‘(SubMnd‘𝑀))‘𝐺))
16	3	submacs 18708	. . . . . . . . . 10 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
17	2, 16	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (SubMnd‘𝑀) ∈ (ACS‘𝐵))
18	17	acsmred 17600	. . . . . . . 8 ⊢ (𝜑 → (SubMnd‘𝑀) ∈ (Moore‘𝐵))
19		mndind.g	. . . . . . . . 9 ⊢ (𝜑 → 𝐺 ⊆ 𝐵)
20		eleq1w 2817	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → (𝑦 ∈ 𝐵 ↔ 𝑎 ∈ 𝐵))
21	20	anbi2d 630	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑎 → (((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑎 ∈ 𝐵)))
22		vex 3479	. . . . . . . . . . . . . . 15 ⊢ 𝑦 ∈ V
23		mndind.ch	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜒))
24	22, 23	sbcie 3821	. . . . . . . . . . . . . 14 ⊢ ([𝑦 / 𝑥]𝜓 ↔ 𝜒)
25		dfsbcq 3780	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → ([𝑦 / 𝑥]𝜓 ↔ [𝑎 / 𝑥]𝜓))
26	24, 25	bitr3id 285	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → (𝜒 ↔ [𝑎 / 𝑥]𝜓))
27		oveq1 7416	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑎 → (𝑦 + 𝑏) = (𝑎 + 𝑏))
28	27	sbceq1d 3783	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑎 → ([(𝑦 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + 𝑏) / 𝑥]𝜓))
29	26, 28	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑎 → ((𝜒 → [(𝑦 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)))
30	21, 29	imbi12d 345	. . . . . . . . . . 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))))
31		eleq1w 2817	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑏 → (𝑧 ∈ 𝐺 ↔ 𝑏 ∈ 𝐺))
32	31	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑏 → ((𝜑 ∧ 𝑧 ∈ 𝐺) ↔ (𝜑 ∧ 𝑏 ∈ 𝐺)))
33	32	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑏 → (((𝜑 ∧ 𝑧 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) ↔ ((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵)))
34		ovex 7442	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 + 𝑧) ∈ V
35		mndind.th	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦 + 𝑧) → (𝜓 ↔ 𝜃))
36	34, 35	sbcie 3821	. . . . . . . . . . . . . . 15 ⊢ ([(𝑦 + 𝑧) / 𝑥]𝜓 ↔ 𝜃)
37		oveq2 7417	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑏 → (𝑦 + 𝑧) = (𝑦 + 𝑏))
38	37	sbceq1d 3783	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑏 → ([(𝑦 + 𝑧) / 𝑥]𝜓 ↔ [(𝑦 + 𝑏) / 𝑥]𝜓))
39	36, 38	bitr3id 285	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑏 → (𝜃 ↔ [(𝑦 + 𝑏) / 𝑥]𝜓))
40	39	imbi2d 341	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑏 → ((𝜒 → 𝜃) ↔ (𝜒 → [(𝑦 + 𝑏) / 𝑥]𝜓)))
41	33, 40	imbi12d 345	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑏 → ((((𝜑 ∧ 𝑧 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) → (𝜒 → 𝜃)) ↔ (((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) → (𝜒 → [(𝑦 + 𝑏) / 𝑥]𝜓))))
42		mndind.i2	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐺) ∧ 𝜒) → 𝜃)
43	42	ex 414	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐺) → (𝜒 → 𝜃))
44	43	3expa 1119	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ 𝐺) → (𝜒 → 𝜃))
45	44	an32s 651	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) → (𝜒 → 𝜃))
46	41, 45	chvarvv 2003	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑦 ∈ 𝐵) → (𝜒 → [(𝑦 + 𝑏) / 𝑥]𝜓))
47	30, 46	chvarvv 2003	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐺) ∧ 𝑎 ∈ 𝐵) → ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓))
48	47	ralrimiva 3147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐺) → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓))
49	19, 48	ssrabdv 4072	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ⊆ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)})
50		mndind.pg	. . . . . . . . 9 ⊢ + = (+_g‘𝑀)
51	3, 50, 4	mndrid 18646	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ Mnd ∧ 𝑎 ∈ 𝐵) → (𝑎 + 0 ) = 𝑎)
52	2, 51	sylan 581	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (𝑎 + 0 ) = 𝑎)
53	52	sbceq1d 3783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ([(𝑎 + 0 ) / 𝑥]𝜓 ↔ [𝑎 / 𝑥]𝜓))
54	53	biimprd 247	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ([𝑎 / 𝑥]𝜓 → [(𝑎 + 0 ) / 𝑥]𝜓))
55	54	ralrimiva 3147	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 0 ) / 𝑥]𝜓))
56		simprrl 780	. . . . . . . . . 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))
57	2	ad2antrr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑀 ∈ Mnd)
58		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
59		simplrl 776	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑐 ∈ 𝐵)
60	3, 50	mndcl 18633	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ Mnd ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵) → (𝑏 + 𝑐) ∈ 𝐵)
61	57, 58, 59, 60	syl3anc 1372	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → (𝑏 + 𝑐) ∈ 𝐵)
62		simpr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) ∧ 𝑎 = (𝑏 + 𝑐)) → 𝑎 = (𝑏 + 𝑐))
63	62	sbceq1d 3783	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) ∧ 𝑎 = (𝑏 + 𝑐)) → ([𝑎 / 𝑥]𝜓 ↔ [(𝑏 + 𝑐) / 𝑥]𝜓))
64		oveq1 7416	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = (𝑏 + 𝑐) → (𝑎 + 𝑑) = ((𝑏 + 𝑐) + 𝑑))
65		simplrr 777	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → 𝑑 ∈ 𝐵)
66	3, 50	mndass 18634	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑀 ∈ Mnd ∧ (𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) → ((𝑏 + 𝑐) + 𝑑) = (𝑏 + (𝑐 + 𝑑)))
67	57, 58, 59, 65, 66	syl13anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) → ((𝑏 + 𝑐) + 𝑑) = (𝑏 + (𝑐 + 𝑑)))
68	64, 67	sylan9eqr 2795	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) ∧ 𝑎 = (𝑏 + 𝑐)) → (𝑎 + 𝑑) = (𝑏 + (𝑐 + 𝑑)))
69	68	sbceq1d 3783	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ (𝑐 ∈ 𝐵 ∧ 𝑑 ∈ 𝐵)) ∧ 𝑏 ∈ 𝐵) ∧ 𝑎 = (𝑏 + 𝑐)) → ([(𝑎 + 𝑑) / 𝑥]𝜓 ↔ [(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓))
70	63, 69	imbi12d 345	. . . . . . . . . . . . . . 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)))
71	61, 70	rspcdv 3605	. . . . . . . . . . . . . 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)))
72	71	ralrimdva 3155	. . . . . . . . . . . . 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)))
73	72	impr 456	. . . . . . . . . . . 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))
74		oveq1 7416	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑎 → (𝑏 + 𝑐) = (𝑎 + 𝑐))
75	74	sbceq1d 3783	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑎 → ([(𝑏 + 𝑐) / 𝑥]𝜓 ↔ [(𝑎 + 𝑐) / 𝑥]𝜓))
76		oveq1 7416	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = 𝑎 → (𝑏 + (𝑐 + 𝑑)) = (𝑎 + (𝑐 + 𝑑)))
77	76	sbceq1d 3783	. . . . . . . . . . . . . 14 ⊢ (𝑏 = 𝑎 → ([(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓 ↔ [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓))
78	75, 77	imbi12d 345	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑎 → (([(𝑏 + 𝑐) / 𝑥]𝜓 → [(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓) ↔ ([(𝑎 + 𝑐) / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
79	78	cbvralvw 3235	. . . . . . . . . . . 12 ⊢ (∀𝑏 ∈ 𝐵 ([(𝑏 + 𝑐) / 𝑥]𝜓 → [(𝑏 + (𝑐 + 𝑑)) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([(𝑎 + 𝑐) / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓))
80	73, 79	sylib 217	. . . . . . . . . . 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))
81	80	adantrrl 723	. . . . . . . . . 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))
82		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)))
83	82	ral2imi 3086	. . . . . . . . . 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)))
84	56, 81, 83	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))
85		oveq2 7417	. . . . . . . . . . . 12 ⊢ (𝑏 = 0 → (𝑎 + 𝑏) = (𝑎 + 0 ))
86	85	sbceq1d 3783	. . . . . . . . . . 11 ⊢ (𝑏 = 0 → ([(𝑎 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + 0 ) / 𝑥]𝜓))
87	86	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑏 = 0 → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + 0 ) / 𝑥]𝜓)))
88	87	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑏 = 0 → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 0 ) / 𝑥]𝜓)))
89		oveq2 7417	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑐 → (𝑎 + 𝑏) = (𝑎 + 𝑐))
90	89	sbceq1d 3783	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑐 → ([(𝑎 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + 𝑐) / 𝑥]𝜓))
91	90	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑏 = 𝑐 → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑐) / 𝑥]𝜓)))
92	91	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑏 = 𝑐 → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑐) / 𝑥]𝜓)))
93		oveq2 7417	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑑 → (𝑎 + 𝑏) = (𝑎 + 𝑑))
94	93	sbceq1d 3783	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑑 → ([(𝑎 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + 𝑑) / 𝑥]𝜓))
95	94	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑏 = 𝑑 → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓)))
96	95	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑏 = 𝑑 → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑑) / 𝑥]𝜓)))
97		oveq2 7417	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑐 + 𝑑) → (𝑎 + 𝑏) = (𝑎 + (𝑐 + 𝑑)))
98	97	sbceq1d 3783	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑐 + 𝑑) → ([(𝑎 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓))
99	98	imbi2d 341	. . . . . . . . . 10 ⊢ (𝑏 = (𝑐 + 𝑑) → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
100	99	ralbidv 3178	. . . . . . . . 9 ⊢ (𝑏 = (𝑐 + 𝑑) → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + (𝑐 + 𝑑)) / 𝑥]𝜓)))
101	3, 50, 4, 2, 55, 84, 88, 92, 96, 100	issubmd 18687	. . . . . . . 8 ⊢ (𝜑 → {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)} ∈ (SubMnd‘𝑀))
102		eqid 2733	. . . . . . . . 9 ⊢ (mrCls‘(SubMnd‘𝑀)) = (mrCls‘(SubMnd‘𝑀))
103	102	mrcsscl 17564	. . . . . . . 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)})
104	18, 49, 101, 103	syl3anc 1372	. . . . . . 7 ⊢ (𝜑 → ((mrCls‘(SubMnd‘𝑀))‘𝐺) ⊆ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)})
105	15, 104	eqsstrd 4021	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)})
106		mndind.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
107	105, 106	sseldd 3984	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)})
108		oveq2 7417	. . . . . . . . . 10 ⊢ (𝑏 = 𝐴 → (𝑎 + 𝑏) = (𝑎 + 𝐴))
109	108	sbceq1d 3783	. . . . . . . . 9 ⊢ (𝑏 = 𝐴 → ([(𝑎 + 𝑏) / 𝑥]𝜓 ↔ [(𝑎 + 𝐴) / 𝑥]𝜓))
110	109	imbi2d 341	. . . . . . . 8 ⊢ (𝑏 = 𝐴 → (([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓)))
111	110	ralbidv 3178	. . . . . . 7 ⊢ (𝑏 = 𝐴 → (∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓) ↔ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓)))
112	111	elrab 3684	. . . . . 6 ⊢ (𝐴 ∈ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)} ↔ (𝐴 ∈ 𝐵 ∧ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓)))
113	112	simprbi 498	. . . . 5 ⊢ (𝐴 ∈ {𝑏 ∈ 𝐵 ∣ ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝑏) / 𝑥]𝜓)} → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓))
114	107, 113	syl 17	. . . 4 ⊢ (𝜑 → ∀𝑎 ∈ 𝐵 ([𝑎 / 𝑥]𝜓 → [(𝑎 + 𝐴) / 𝑥]𝜓))
115	14, 114, 6	rspcdva 3614	. . 3 ⊢ (𝜑 → ([ 0 / 𝑥]𝜓 → [( 0 + 𝐴) / 𝑥]𝜓))
116	10, 115	mpd 15	. 2 ⊢ (𝜑 → [( 0 + 𝐴) / 𝑥]𝜓)
117	3, 50, 4	mndlid 18645	. . . . 5 ⊢ ((𝑀 ∈ Mnd ∧ 𝐴 ∈ 𝐵) → ( 0 + 𝐴) = 𝐴)
118	2, 106, 117	syl2anc 585	. . . 4 ⊢ (𝜑 → ( 0 + 𝐴) = 𝐴)
119	118	sbceq1d 3783	. . 3 ⊢ (𝜑 → ([( 0 + 𝐴) / 𝑥]𝜓 ↔ [𝐴 / 𝑥]𝜓))
120		mndind.et	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜓 ↔ 𝜂))
121	120	sbcieg 3818	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ([𝐴 / 𝑥]𝜓 ↔ 𝜂))
122	106, 121	syl 17	. . 3 ⊢ (𝜑 → ([𝐴 / 𝑥]𝜓 ↔ 𝜂))
123	119, 122	bitrd 279	. 2 ⊢ (𝜑 → ([( 0 + 𝐴) / 𝑥]𝜓 ↔ 𝜂))
124	116, 123	mpbid 231	1 ⊢ (𝜑 → 𝜂)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 [wsbc 3778 ⊆ wss 3949 ‘cfv 6544 (class class class)co 7409 Basecbs 17144 +_gcplusg 17197 0_gc0g 17385 Moorecmre 17526 mrClscmrc 17527 ACScacs 17529 Mndcmnd 18625 SubMndcsubmnd 18670

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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-om 7856 df-1o 8466 df-en 8940 df-fin 8943 df-0g 17387 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672

This theorem is referenced by: mdetunilem7 22120

W3C validator