![]() |
Mathbox for Jeff Madsen |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > rngorn1 | Structured version Visualization version GIF version |
Description: In a unital ring the range 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 |
---|---|
rngorn1 | ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rnplrnml0.2 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngogrpo 34666 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | grporndm 27966 | . . 3 ⊢ (𝐺 ∈ GrpOp → ran 𝐺 = dom dom 𝐺) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐺) |
5 | rnplrnml0.1 | . . 3 ⊢ 𝐻 = (2nd ‘𝑅) | |
6 | 5, 1 | rngodm1dm2 34688 | . 2 ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻) |
7 | 4, 6 | eqtrd 2829 | 1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1520 ∈ wcel 2079 dom cdm 5435 ran crn 5436 ‘cfv 6217 1st c1st 7534 2nd c2nd 7535 GrpOpcgr 27945 RingOpscrngo 34650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-13 2342 ax-ext 2767 ax-sep 5088 ax-nul 5095 ax-pow 5150 ax-pr 5214 ax-un 7310 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1080 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-mo 2574 df-eu 2610 df-clab 2774 df-cleq 2786 df-clel 2861 df-nfc 2933 df-ne 2983 df-ral 3108 df-rex 3109 df-rab 3112 df-v 3434 df-sbc 3702 df-csb 3807 df-dif 3857 df-un 3859 df-in 3861 df-ss 3869 df-nul 4207 df-if 4376 df-sn 4467 df-pr 4469 df-op 4473 df-uni 4740 df-iun 4821 df-br 4957 df-opab 5019 df-mpt 5036 df-id 5340 df-xp 5441 df-rel 5442 df-cnv 5443 df-co 5444 df-dm 5445 df-rn 5446 df-iota 6181 df-fun 6219 df-fn 6220 df-f 6221 df-fo 6223 df-fv 6225 df-ov 7010 df-1st 7536 df-2nd 7537 df-grpo 27949 df-ablo 28001 df-rngo 34651 |
This theorem is referenced by: rngomndo 34691 |
Copyright terms: Public domain | W3C validator |