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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngodm1dm2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
rnplrnml0.1 | ⊢ 𝐻 = (2nd ‘𝑅) |
rnplrnml0.2 | ⊢ 𝐺 = (1st ‘𝑅) |
Ref | Expression |
---|---|
rngodm1dm2 | ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻) |
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) |
Colors of variables: wff setvar class |
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 1st c1st 7499 2nd 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 |
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