Theorem caovimo 5714

Description: Uniqueness of inverse element in commutative, associative operation with identity. The identity element is 𝐵. (Contributed by Jim Kingdon, 18-Sep-2019.)

Hypotheses

Ref	Expression
caovimo.idel	⊢ 𝐵 ∈ 𝑆
caovimo.com	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
caovimo.ass	⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
caovimo.id	⊢ (𝑥 ∈ 𝑆 → (𝑥𝐹𝐵) = 𝑥)

Assertion

Ref	Expression
caovimo	⊢ (𝐴 ∈ 𝑆 → ∃*𝑤(𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))

Distinct variable groups:   𝑤,𝐴,𝑥,𝑦,𝑧   𝑤,𝐵,𝑥,𝑦   𝑤,𝐹,𝑥,𝑦,𝑧   𝑤,𝑆,𝑥,𝑦,𝑧

Allowed substitution hint:   𝐵(𝑧)

Proof of Theorem caovimo

Dummy variable 𝑣 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 5539	. . . . . . 7 ⊢ ((𝐴𝐹𝑤) = 𝐵 → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
2	1	adantl 271	. . . . . 6 ⊢ ((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
3	2	3ad2ant2 960	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
4		df-3an 921	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ↔ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 ∈ 𝑆))
5		caovimo.ass	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
6	5	adantl 271	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
7		simp1 938	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → 𝐴 ∈ 𝑆)
8		simp2 939	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → 𝑤 ∈ 𝑆)
9		simp3 940	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
10	6, 7, 8, 9	caovassd 5680	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐴𝐹(𝑤𝐹𝑣)))
11		caovimo.com	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
12	11	adantl 271	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
13	7, 8, 9, 12, 6	caov12d 5702	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝐴𝐹(𝑤𝐹𝑣)) = (𝑤𝐹(𝐴𝐹𝑣)))
14	10, 13	eqtrd 2113	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝐴𝐹𝑣)))
15	14	adantr 270	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝐴𝐹𝑣)))
16		oveq2 5540	. . . . . . . . . . . 12 ⊢ ((𝐴𝐹𝑣) = 𝐵 → (𝑤𝐹(𝐴𝐹𝑣)) = (𝑤𝐹𝐵))
17		oveq1 5539	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑥𝐹𝐵) = (𝑤𝐹𝐵))
18		id 19	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
19	17, 18	eqeq12d 2095	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑤𝐹𝐵) = 𝑤))
20		caovimo.id	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑆 → (𝑥𝐹𝐵) = 𝑥)
21	19, 20	vtoclga 2664	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑆 → (𝑤𝐹𝐵) = 𝑤)
22	16, 21	sylan9eqr 2135	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵) → (𝑤𝐹(𝐴𝐹𝑣)) = 𝑤)
23	22	3ad2antl2 1101	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → (𝑤𝐹(𝐴𝐹𝑣)) = 𝑤)
24	15, 23	eqtrd 2113	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
25	4, 24	sylanbr 279	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 ∈ 𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
26	25	anasss 391	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
27	26	3impa 1133	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
28	27	3adant2r 1164	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
29	11	adantl 271	. . . . . . . . 9 ⊢ ((𝑣 ∈ 𝑆 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
30		caovimo.idel	. . . . . . . . . 10 ⊢ 𝐵 ∈ 𝑆
31	30	a1i 9	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝑆 → 𝐵 ∈ 𝑆)
32		id 19	. . . . . . . . 9 ⊢ (𝑣 ∈ 𝑆 → 𝑣 ∈ 𝑆)
33	29, 31, 32	caovcomd 5677	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑆 → (𝐵𝐹𝑣) = (𝑣𝐹𝐵))
34		oveq1 5539	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑥𝐹𝐵) = (𝑣𝐹𝐵))
35		id 19	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → 𝑥 = 𝑣)
36	34, 35	eqeq12d 2095	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑣𝐹𝐵) = 𝑣))
37	36, 20	vtoclga 2664	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑆 → (𝑣𝐹𝐵) = 𝑣)
38	33, 37	eqtrd 2113	. . . . . . 7 ⊢ (𝑣 ∈ 𝑆 → (𝐵𝐹𝑣) = 𝑣)
39	38	adantr 270	. . . . . 6 ⊢ ((𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵) → (𝐵𝐹𝑣) = 𝑣)
40	39	3ad2ant3 961	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → (𝐵𝐹𝑣) = 𝑣)
41	3, 28, 40	3eqtr3d 2121	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣)
42	41	3expib 1141	. . 3 ⊢ (𝐴 ∈ 𝑆 → (((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
43	42	alrimivv 1796	. 2 ⊢ (𝐴 ∈ 𝑆 → ∀𝑤∀𝑣(((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
44		eleq1 2141	. . . 4 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑆 ↔ 𝑣 ∈ 𝑆))
45		oveq2 5540	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝐴𝐹𝑤) = (𝐴𝐹𝑣))
46	45	eqeq1d 2089	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝐴𝐹𝑤) = 𝐵 ↔ (𝐴𝐹𝑣) = 𝐵))
47	44, 46	anbi12d 456	. . 3 ⊢ (𝑤 = 𝑣 → ((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)))
48	47	mo4 2002	. 2 ⊢ (∃*𝑤(𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ ∀𝑤∀𝑣(((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
49	43, 48	sylibr 132	1 ⊢ (𝐴 ∈ 𝑆 → ∃*𝑤(𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 102   ∧ w3a 919  ∀wal 1282   = wceq 1284   ∈ wcel 1433  ∃*wmo 1942  (class class class)co 5532

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-fv 4930  df-ov 5535

This theorem is referenced by:  recmulnqg  6581