Theorem rngodm1dm2 37892

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 37870	. . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp)
3		eqid 2740	. . . 4 ⊢ ran 𝐺 = ran 𝐺
4	3	grpofo 30531	. . 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 37860	. 2 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
8		fof 6834	. . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺)
9	8	fdmd 6757	. . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺))
10		fdm 6756	. . . 4 ⊢ (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom 𝐻 = (ran 𝐺 × ran 𝐺))
11		eqtr 2763	. . . . . . 7 ⊢ ((dom 𝐺 = (ran 𝐺 × ran 𝐺) ∧ (ran 𝐺 × ran 𝐺) = dom 𝐻) → dom 𝐺 = dom 𝐻)
12	11	dmeqd 5930	. . . . . 6 ⊢ ((dom 𝐺 = (ran 𝐺 × ran 𝐺) ∧ (ran 𝐺 × ran 𝐺) = dom 𝐻) → dom dom 𝐺 = dom dom 𝐻)
13	12	expcom 413	. . . . 5 ⊢ ((ran 𝐺 × ran 𝐺) = dom 𝐻 → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻))
14	13	eqcoms 2748	. . . 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 395   = wceq 1537   ∈ wcel 2108   × cxp 5698  dom cdm 5700  ran crn 5701  ⟶wf 6569  –onto→wfo 6571  ‘cfv 6573  1^st c1st 8028  2^nd c2nd 8029  GrpOpcgr 30521  RingOpscrngo 37854

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-ov 7451  df-1st 8030  df-2nd 8031  df-grpo 30525  df-ablo 30577  df-rngo 37855

This theorem is referenced by:  rngorn1  37893