Theorem ring2 8145

Description: A ring element plus itself is two times the element. (Contributed by Steve Rodriguez, 9-Sep-2007.)

Hypotheses

Ref	Expression
ring2.1	⊢ G = (1st ‘R)
ring2.2	⊢ H = (2nd ‘R)
ring2.3	⊢ X = ran G

Assertion

Ref	Expression
ring2	⊢ ((R ∈ Ring ⋀ A ∈ X) → ∃x ∈ X (AGA) = ((xGx)HA))

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

Proof of Theorem ring2

Step	Hyp	Ref	Expression
1		ring2.1	. . . 4 ⊢ G = (1st ‘R)
2		ring2.2	. . . 4 ⊢ H = (2nd ‘R)
3		ring2.3	. . . 4 ⊢ X = ran G
4	1, 2, 3	ringid 8141	. . 3 ⊢ ((R ∈ Ring ⋀ A ∈ X) → ∃x ∈ X ((AHx) = A ⋀ (xHA) = A))
5		pm3.27 323	. . . 4 ⊢ (((AHx) = A ⋀ (xHA) = A) → (xHA) = A)
6	5	r19.22si 1737	. . 3 ⊢ (∃x ∈ X ((AHx) = A ⋀ (xHA) = A) → ∃x ∈ X (xHA) = A)
7		opreq12 3976	. . . . 5 ⊢ (((xHA) = A ⋀ (xHA) = A) → ((xHA)G(xHA)) = (AGA))
8	7	anidms 436	. . . 4 ⊢ ((xHA) = A → ((xHA)G(xHA)) = (AGA))
9	8	r19.22si 1737	. . 3 ⊢ (∃x ∈ X (xHA) = A → ∃x ∈ X ((xHA)G(xHA)) = (AGA))
10	4, 6, 9	3syl 20	. 2 ⊢ ((R ∈ Ring ⋀ A ∈ X) → ∃x ∈ X ((xHA)G(xHA)) = (AGA))
11		eqtrt 1495	. . . . . . 7 ⊢ ((((xGx)HA) = ((xHA)G(xHA)) ⋀ ((xHA)G(xHA)) = (AGA)) → ((xGx)HA) = (AGA))
12	11	eqcomd 1483	. . . . . 6 ⊢ ((((xGx)HA) = ((xHA)G(xHA)) ⋀ ((xHA)G(xHA)) = (AGA)) → (AGA) = ((xGx)HA))
13	1, 2, 3	ringdir 8143	. . . . . . . . . . 11 ⊢ ((R ∈ Ring ⋀ (x ∈ X ⋀ x ∈ X ⋀ A ∈ X)) → ((xGx)HA) = ((xHA)G(xHA)))
14	13	expcom 374	. . . . . . . . . 10 ⊢ ((x ∈ X ⋀ x ∈ X ⋀ A ∈ X) → (R ∈ Ring → ((xGx)HA) = ((xHA)G(xHA))))
15	14	3expia 837	. . . . . . . . 9 ⊢ ((x ∈ X ⋀ x ∈ X) → (A ∈ X → (R ∈ Ring → ((xGx)HA) = ((xHA)G(xHA)))))
16	15	anidms 436	. . . . . . . 8 ⊢ (x ∈ X → (A ∈ X → (R ∈ Ring → ((xGx)HA) = ((xHA)G(xHA)))))
17	16	3imp 829	. . . . . . 7 ⊢ ((x ∈ X ⋀ A ∈ X ⋀ R ∈ Ring) → ((xGx)HA) = ((xHA)G(xHA)))
18	17	3com13 840	. . . . . 6 ⊢ ((R ∈ Ring ⋀ A ∈ X ⋀ x ∈ X) → ((xGx)HA) = ((xHA)G(xHA)))
19	12, 18	sylan 450	. . . . 5 ⊢ (((R ∈ Ring ⋀ A ∈ X ⋀ x ∈ X) ⋀ ((xHA)G(xHA)) = (AGA)) → (AGA) = ((xGx)HA))
20	19	ex 373	. . . 4 ⊢ ((R ∈ Ring ⋀ A ∈ X ⋀ x ∈ X) → (((xHA)G(xHA)) = (AGA) → (AGA) = ((xGx)HA)))
21	20	3expa 835	. . 3 ⊢ (((R ∈ Ring ⋀ A ∈ X) ⋀ x ∈ X) → (((xHA)G(xHA)) = (AGA) → (AGA) = ((xGx)HA)))
22	21	r19.22dva 1742	. 2 ⊢ ((R ∈ Ring ⋀ A ∈ X) → (∃x ∈ X ((xHA)G(xHA)) = (AGA) → ∃x ∈ X (AGA) = ((xGx)HA)))
23	10, 22	mpd 26	1 ⊢ ((R ∈ Ring ⋀ A ∈ X) → ∃x ∈ X (AGA) = ((xGx)HA))

Syntax hints:   → wi 3   ⋀ wa 223   ⋀ w3a 777   = wceq 958   ∈ wcel 960  ∃wrex 1649  ran crn 3177   ‘cfv 3188  (class class class)co 3969  1^st c1st 4083  2^nd c2nd 4084  Ringcring 8135

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

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