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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 36065 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | grporndm 28869 | . . 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 36087 | . 2 ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻) |
7 | 4, 6 | eqtrd 2778 | 1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 dom cdm 5591 ran crn 5592 ‘cfv 6435 1st c1st 7829 2nd c2nd 7830 GrpOpcgr 28848 RingOpscrngo 36049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5225 ax-nul 5232 ax-pr 5354 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-id 5491 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-fo 6441 df-fv 6443 df-ov 7280 df-1st 7831 df-2nd 7832 df-grpo 28852 df-ablo 28904 df-rngo 36050 |
This theorem is referenced by: rngomndo 36090 |
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