Theorem rngodm1dm2 34649

Description: In a unital ring the domain of the first variable of the addition equals the domain of the first variable of the multiplication. (Contributed by FL, 24-Jan-2010.) (New usage is discouraged.)

Hypotheses

Ref	Expression
rnplrnml0.1	⊢ 𝐻 = (2nd ‘𝑅)
rnplrnml0.2	⊢ 𝐺 = (1st ‘𝑅)

Assertion

Ref	Expression
rngodm1dm2	⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)

Proof of Theorem rngodm1dm2

Step	Hyp	Ref	Expression
1		rnplrnml0.2	. . . 4 ⊢ 𝐺 = (1st ‘𝑅)
2	1	rngogrpo 34627	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		eqid 2779	. . . 4 ⊢ ran 𝐺 = ran 𝐺
4	3	grpofo 28053	. . 3 ⊢ (𝐺 ∈ GrpOp → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
5	2, 4	syl 17	. 2 ⊢ (𝑅 ∈ RingOps → 𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺)
6		rnplrnml0.1	. . 3 ⊢ 𝐻 = (2nd ‘𝑅)
7	1, 6, 3	rngosm 34617	. 2 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
8		fof 6419	. . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
9	8	fdmd 6353	. . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
10		fdm 6352	. . . 4 ⊢ (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom 𝐻 = (ran 𝐺 × ran 𝐺))
11		eqtr 2800	. . . . . . 7 ⊢ ((dom 𝐺 = (ran 𝐺 × ran 𝐺) ∧ (ran 𝐺 × ran 𝐺) = dom 𝐻) → dom 𝐺 = dom 𝐻)
12	11	dmeqd 5624	. . . . . 6 ⊢ ((dom 𝐺 = (ran 𝐺 × ran 𝐺) ∧ (ran 𝐺 × ran 𝐺) = dom 𝐻) → dom dom 𝐺 = dom dom 𝐻)
13	12	expcom 406	. . . . 5 ⊢ ((ran 𝐺 × ran 𝐺) = dom 𝐻 → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻))
14	13	eqcoms 2787	. . . 4 ⊢ (dom 𝐻 = (ran 𝐺 × ran 𝐺) → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻))
15	10, 14	syl 17	. . 3 ⊢ (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻))
16	9, 15	syl5com 31	. 2 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom dom 𝐺 = dom dom 𝐻))
17	5, 7, 16	sylc 65	1 ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻)

Syntax hints:   → wi 4   ∧ wa 387   = wceq 1507   ∈ wcel 2050   × cxp 5405  dom cdm 5407  ran crn 5408  ⟶wf 6184  –onto→wfo 6186  ‘cfv 6188  1^st c1st 7499  2^nd c2nd 7500  GrpOpcgr 28043  RingOpscrngo 34611

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2751  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-sbc 3683  df-csb 3788  df-dif 3833  df-un 3835  df-in 3837  df-ss 3844  df-nul 4180  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fo 6194  df-fv 6196  df-ov 6979  df-1st 7501  df-2nd 7502  df-grpo 28047  df-ablo 28099  df-rngo 34612

This theorem is referenced by:  rngorn1  34650