Theorem grpinvalem 18639

Description: Lemma for grpinva 18640. (Contributed by NM, 9-Aug-2013.)

Hypotheses

Ref	Expression
grpinva.c	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
grpinva.o	⊢ (𝜑 → 𝑂 ∈ 𝐵)
grpinva.i	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
grpinva.a	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
grpinva.r	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)
grpinvalem.x	⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)
grpinvalem.e	⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑋) = 𝑋)

Assertion

Ref	Expression
grpinvalem	⊢ ((𝜑 ∧ 𝜓) → 𝑋 = 𝑂)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑂,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧   𝑥, + ,𝑦,𝑧   𝑦,𝑋,𝑧   𝜓,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑧)   𝑋(𝑥)

Proof of Theorem grpinvalem

Dummy variables 𝑢 𝑣 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpinva.r	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)
2	1	ralrimiva 3132	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂)
3		oveq2 7371	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑦 + 𝑥) = (𝑦 + 𝑧))
4	3	eqeq1d 2742	. . . . . 6 ⊢ (𝑥 = 𝑧 → ((𝑦 + 𝑥) = 𝑂 ↔ (𝑦 + 𝑧) = 𝑂))
5	4	rexbidv 3164	. . . . 5 ⊢ (𝑥 = 𝑧 → (∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑧) = 𝑂))
6	5	cbvralvw 3218	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 𝑂 ↔ ∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑧) = 𝑂)
7	2, 6	sylib 219	. . 3 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑧) = 𝑂)
8		grpinvalem.x	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 ∈ 𝐵)
9		oveq2 7371	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝑦 + 𝑧) = (𝑦 + 𝑋))
10	9	eqeq1d 2742	. . . . 5 ⊢ (𝑧 = 𝑋 → ((𝑦 + 𝑧) = 𝑂 ↔ (𝑦 + 𝑋) = 𝑂))
11	10	rexbidv 3164	. . . 4 ⊢ (𝑧 = 𝑋 → (∃𝑦 ∈ 𝐵 (𝑦 + 𝑧) = 𝑂 ↔ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 𝑂))
12	11	rspccva 3566	. . 3 ⊢ ((∀𝑧 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑧) = 𝑂 ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 𝑂)
13	7, 8, 12	syl2an2r 691	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 𝑂)
14		grpinvalem.e	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑋 + 𝑋) = 𝑋)
15	14	oveq2d 7379	. . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
16	15	adantr 481	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = (𝑦 + 𝑋))
17		simprr 778	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + 𝑋) = 𝑂)
18	17	oveq1d 7378	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑂 + 𝑋))
19		grpinva.a	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐵)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
20	19	caovassg 7561	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
21	20	ad4ant14 758	. . . . 5 ⊢ ((((𝜑 ∧ 𝜓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) ∧ (𝑢 ∈ 𝐵 ∧ 𝑣 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → ((𝑢 + 𝑣) + 𝑤) = (𝑢 + (𝑣 + 𝑤)))
22		simprl 776	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑦 ∈ 𝐵)
23	8	adantr 481	. . . . 5 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋 ∈ 𝐵)
24	21, 22, 23, 23	caovassd 7562	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → ((𝑦 + 𝑋) + 𝑋) = (𝑦 + (𝑋 + 𝑋)))
25		oveq2 7371	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑂 + 𝑦) = (𝑂 + 𝑋))
26		id 22	. . . . . . 7 ⊢ (𝑦 = 𝑋 → 𝑦 = 𝑋)
27	25, 26	eqeq12d 2756	. . . . . 6 ⊢ (𝑦 = 𝑋 → ((𝑂 + 𝑦) = 𝑦 ↔ (𝑂 + 𝑋) = 𝑋))
28		grpinva.i	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑂 + 𝑥) = 𝑥)
29	28	ralrimiva 3132	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 (𝑂 + 𝑥) = 𝑥)
30		oveq2 7371	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑂 + 𝑥) = (𝑂 + 𝑦))
31		id 22	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
32	30, 31	eqeq12d 2756	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝑂 + 𝑥) = 𝑥 ↔ (𝑂 + 𝑦) = 𝑦))
33	32	cbvralvw 3218	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 (𝑂 + 𝑥) = 𝑥 ↔ ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
34	29, 33	sylib 219	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
35	34	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑦 ∈ 𝐵 (𝑂 + 𝑦) = 𝑦)
36	27, 35, 8	rspcdva 3568	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (𝑂 + 𝑋) = 𝑋)
37	36	adantr 481	. . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑂 + 𝑋) = 𝑋)
38	18, 24, 37	3eqtr3d 2783	. . 3 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → (𝑦 + (𝑋 + 𝑋)) = 𝑋)
39	16, 38, 17	3eqtr3d 2783	. 2 ⊢ (((𝜑 ∧ 𝜓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑦 + 𝑋) = 𝑂)) → 𝑋 = 𝑂)
40	13, 39	rexlimddv 3147	1 ⊢ ((𝜑 ∧ 𝜓) → 𝑋 = 𝑂)

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  (class class class)co 7363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366

This theorem is referenced by:  grpinva  18640