Theorem ringid 8141

Description: The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Hypotheses

Ref	Expression
ringid.1	⊢ G = (1st ‘R)
ringid.2	⊢ H = (2nd ‘R)
ringid.3	⊢ X = ran G

Assertion

Ref	Expression
ringid	⊢ ((R ∈ Ring ⋀ A ∈ X) → ∃x ∈ X ((AHx) = A ⋀ (xHA) = A))

Distinct variable groups:   x,A   x,G   x,H   x,X

Proof of Theorem ringid

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . . . . 7 ⊢ (y = A → (yHx) = (AHx))
2		id 59	. . . . . . 7 ⊢ (y = A → y = A)
3	1, 2	eqeq12d 1492	. . . . . 6 ⊢ (y = A → ((yHx) = y ↔ (AHx) = A))
4		opreq2 3975	. . . . . . 7 ⊢ (y = A → (xHy) = (xHA))
5	4, 2	eqeq12d 1492	. . . . . 6 ⊢ (y = A → ((xHy) = y ↔ (xHA) = A))
6	3, 5	anbi12d 630	. . . . 5 ⊢ (y = A → (((yHx) = y ⋀ (xHy) = y) ↔ ((AHx) = A ⋀ (xHA) = A)))
7	6	rexbidv 1667	. . . 4 ⊢ (y = A → (∃x ∈ X ((yHx) = y ⋀ (xHy) = y) ↔ ∃x ∈ X ((AHx) = A ⋀ (xHA) = A)))
8	7	imbi2d 614	. . 3 ⊢ (y = A → ((R ∈ Ring → ∃x ∈ X ((yHx) = y ⋀ (xHy) = y)) ↔ (R ∈ Ring → ∃x ∈ X ((AHx) = A ⋀ (xHA) = A))))
9		ringid.1	. . . . . . . 8 ⊢ G = (1st ‘R)
10		ringid.2	. . . . . . . 8 ⊢ H = (2nd ‘R)
11		ringid.3	. . . . . . . 8 ⊢ X = ran G
12	9, 10, 11	ringi 8138	. . . . . . 7 ⊢ (R ∈ Ring → ((G ∈ Abel ⋀ H:(X × X)–→X) ⋀ (∀x ∈ X ∀y ∈ X ∀z ∈ X (((xHy)Hz) = (xH(yHz)) ⋀ (xH(yGz)) = ((xHy)G(xHz)) ⋀ ((xGy)Hz) = ((xHz)G(yHz))) ⋀ ∃x ∈ X ∀y ∈ X ((yHx) = y ⋀ (xHy) = y))))
13	12	pm3.27d 325	. . . . . 6 ⊢ (R ∈ Ring → (∀x ∈ X ∀y ∈ X ∀z ∈ X (((xHy)Hz) = (xH(yHz)) ⋀ (xH(yGz)) = ((xHy)G(xHz)) ⋀ ((xGy)Hz) = ((xHz)G(yHz))) ⋀ ∃x ∈ X ∀y ∈ X ((yHx) = y ⋀ (xHy) = y)))
14	13	pm3.27d 325	. . . . 5 ⊢ (R ∈ Ring → ∃x ∈ X ∀y ∈ X ((yHx) = y ⋀ (xHy) = y))
15		r19.12 1743	. . . . 5 ⊢ (∃x ∈ X ∀y ∈ X ((yHx) = y ⋀ (xHy) = y) → ∀y ∈ X ∃x ∈ X ((yHx) = y ⋀ (xHy) = y))
16		ra4 1697	. . . . 5 ⊢ (∀y ∈ X ∃x ∈ X ((yHx) = y ⋀ (xHy) = y) → (y ∈ X → ∃x ∈ X ((yHx) = y ⋀ (xHy) = y)))
17	14, 15, 16	3syl 20	. . . 4 ⊢ (R ∈ Ring → (y ∈ X → ∃x ∈ X ((yHx) = y ⋀ (xHy) = y)))
18	17	com12 11	. . 3 ⊢ (y ∈ X → (R ∈ Ring → ∃x ∈ X ((yHx) = y ⋀ (xHy) = y)))
19	8, 18	vtoclga 1855	. 2 ⊢ (A ∈ X → (R ∈ Ring → ∃x ∈ X ((AHx) = A ⋀ (xHA) = A)))
20	19	impcom 351	1 ⊢ ((R ∈ Ring ⋀ A ∈ X) → ∃x ∈ X ((AHx) = A ⋀ (xHA) = A))

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960  ∀wral 1648  ∃wrex 1649   × cxp 3174  ran crn 3177  –→wf 3184   ‘cfv 3188  (class class class)co 3969  1^st c1st 4083  2^nd c2nd 4084  Abelcabl 8095  Ringcring 8135

This theorem is referenced by:  ring2 8145

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fv 3204  df-opr 3971  df-1st 4085  df-2nd 4086  df-ring 8136

Copyright terms: Public domain