Theorem caovimo 6081

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 5895	. . . . . . 7 ⊢ ((𝐴𝐹𝑤) = 𝐵 → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
2	1	adantl 277	. . . . . 6 ⊢ ((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
3	2	3ad2ant2 1020	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐵𝐹𝑣))
4		df-3an 981	. . . . . . . . 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 277	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥𝐹𝑦)𝐹𝑧) = (𝑥𝐹(𝑦𝐹𝑧)))
7		simp1 998	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → 𝐴 ∈ 𝑆)
8		simp2 999	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → 𝑤 ∈ 𝑆)
9		simp3 1000	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
10	6, 7, 8, 9	caovassd 6047	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝐴𝐹(𝑤𝐹𝑣)))
11		caovimo.com	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
12	11	adantl 277	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥))
13	7, 8, 9, 12, 6	caov12d 6069	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → (𝐴𝐹(𝑤𝐹𝑣)) = (𝑤𝐹(𝐴𝐹𝑣)))
14	10, 13	eqtrd 2220	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝐴𝐹𝑣)))
15	14	adantr 276	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = (𝑤𝐹(𝐴𝐹𝑣)))
16		oveq2 5896	. . . . . . . . . . . 12 ⊢ ((𝐴𝐹𝑣) = 𝐵 → (𝑤𝐹(𝐴𝐹𝑣)) = (𝑤𝐹𝐵))
17		oveq1 5895	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → (𝑥𝐹𝐵) = (𝑤𝐹𝐵))
18		id 19	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
19	17, 18	eqeq12d 2202	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑤 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑤𝐹𝐵) = 𝑤))
20		caovimo.id	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ 𝑆 → (𝑥𝐹𝐵) = 𝑥)
21	19, 20	vtoclga 2815	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ 𝑆 → (𝑤𝐹𝐵) = 𝑤)
22	16, 21	sylan9eqr 2242	. . . . . . . . . . 11 ⊢ ((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵) → (𝑤𝐹(𝐴𝐹𝑣)) = 𝑤)
23	22	3ad2antl2 1161	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → (𝑤𝐹(𝐴𝐹𝑣)) = 𝑤)
24	15, 23	eqtrd 2220	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ 𝑣 ∈ 𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
25	4, 24	sylanbr 285	. . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ 𝑣 ∈ 𝑆) ∧ (𝐴𝐹𝑣) = 𝐵) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
26	25	anasss 399	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
27	26	3impa 1195	. . . . . 6 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝑤 ∈ 𝑆 ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
28	27	3adant2r 1234	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → ((𝐴𝐹𝑤)𝐹𝑣) = 𝑤)
29	11	adantl 277	. . . . . . . . 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 6044	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑆 → (𝐵𝐹𝑣) = (𝑣𝐹𝐵))
34		oveq1 5895	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑥𝐹𝐵) = (𝑣𝐹𝐵))
35		id 19	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → 𝑥 = 𝑣)
36	34, 35	eqeq12d 2202	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((𝑥𝐹𝐵) = 𝑥 ↔ (𝑣𝐹𝐵) = 𝑣))
37	36, 20	vtoclga 2815	. . . . . . . 8 ⊢ (𝑣 ∈ 𝑆 → (𝑣𝐹𝐵) = 𝑣)
38	33, 37	eqtrd 2220	. . . . . . 7 ⊢ (𝑣 ∈ 𝑆 → (𝐵𝐹𝑣) = 𝑣)
39	38	adantr 276	. . . . . 6 ⊢ ((𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵) → (𝐵𝐹𝑣) = 𝑣)
40	39	3ad2ant3 1021	. . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → (𝐵𝐹𝑣) = 𝑣)
41	3, 28, 40	3eqtr3d 2228	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣)
42	41	3expib 1207	. . 3 ⊢ (𝐴 ∈ 𝑆 → (((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
43	42	alrimivv 1885	. 2 ⊢ (𝐴 ∈ 𝑆 → ∀𝑤∀𝑣(((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
44		eleq1 2250	. . . 4 ⊢ (𝑤 = 𝑣 → (𝑤 ∈ 𝑆 ↔ 𝑣 ∈ 𝑆))
45		oveq2 5896	. . . . 5 ⊢ (𝑤 = 𝑣 → (𝐴𝐹𝑤) = (𝐴𝐹𝑣))
46	45	eqeq1d 2196	. . . 4 ⊢ (𝑤 = 𝑣 → ((𝐴𝐹𝑤) = 𝐵 ↔ (𝐴𝐹𝑣) = 𝐵))
47	44, 46	anbi12d 473	. . 3 ⊢ (𝑤 = 𝑣 → ((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)))
48	47	mo4 2097	. 2 ⊢ (∃*𝑤(𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ↔ ∀𝑤∀𝑣(((𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵) ∧ (𝑣 ∈ 𝑆 ∧ (𝐴𝐹𝑣) = 𝐵)) → 𝑤 = 𝑣))
49	43, 48	sylibr 134	1 ⊢ (𝐴 ∈ 𝑆 → ∃*𝑤(𝑤 ∈ 𝑆 ∧ (𝐴𝐹𝑤) = 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 979  ∀wal 1361   = wceq 1363  ∃*wmo 2037   ∈ wcel 2158  (class class class)co 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-v 2751  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-iota 5190  df-fv 5236  df-ov 5891

This theorem is referenced by:  recmulnqg  7403