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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 37872 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | grporndm 30544 | . . 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 37894 | . 2 ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻) |
7 | 4, 6 | eqtrd 2780 | 1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = dom dom 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2108 dom cdm 5700 ran crn 5701 ‘cfv 6575 1st c1st 8030 2nd c2nd 8031 GrpOpcgr 30523 RingOpscrngo 37856 |
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 7772 |
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 6527 df-fun 6577 df-fn 6578 df-f 6579 df-fo 6581 df-fv 6583 df-ov 7453 df-1st 8032 df-2nd 8033 df-grpo 30527 df-ablo 30579 df-rngo 37857 |
This theorem is referenced by: rngomndo 37897 |
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