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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 32879 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
3 | eqid 2610 | . . . 4 ⊢ ran 𝐺 = ran 𝐺 | |
4 | 3 | grpofo 26737 | . . 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 32869 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) |
8 | fof 6028 | . . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → 𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) | |
9 | fdm 5964 | . . . 4 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺)) | |
10 | 8, 9 | syl 17 | . . 3 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → dom 𝐺 = (ran 𝐺 × ran 𝐺)) |
11 | fdm 5964 | . . . 4 ⊢ (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom 𝐻 = (ran 𝐺 × ran 𝐺)) | |
12 | eqtr 2629 | . . . . . . 7 ⊢ ((dom 𝐺 = (ran 𝐺 × ran 𝐺) ∧ (ran 𝐺 × ran 𝐺) = dom 𝐻) → dom 𝐺 = dom 𝐻) | |
13 | 12 | dmeqd 5248 | . . . . . 6 ⊢ ((dom 𝐺 = (ran 𝐺 × ran 𝐺) ∧ (ran 𝐺 × ran 𝐺) = dom 𝐻) → dom dom 𝐺 = dom dom 𝐻) |
14 | 13 | expcom 450 | . . . . 5 ⊢ ((ran 𝐺 × ran 𝐺) = dom 𝐻 → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻)) |
15 | 14 | eqcoms 2618 | . . . 4 ⊢ (dom 𝐻 = (ran 𝐺 × ran 𝐺) → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻)) |
16 | 11, 15 | syl 17 | . . 3 ⊢ (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → (dom 𝐺 = (ran 𝐺 × ran 𝐺) → dom dom 𝐺 = dom dom 𝐻)) |
17 | 10, 16 | syl5com 31 | . 2 ⊢ (𝐺:(ran 𝐺 × ran 𝐺)–onto→ran 𝐺 → (𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 → dom dom 𝐺 = dom dom 𝐻)) |
18 | 5, 7, 17 | sylc 63 | 1 ⊢ (𝑅 ∈ RingOps → dom dom 𝐺 = dom dom 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 × cxp 5036 dom cdm 5038 ran crn 5039 ⟶wf 5800 –onto→wfo 5802 ‘cfv 5804 1st c1st 7057 2nd c2nd 7058 GrpOpcgr 26727 RingOpscrngo 32863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-ov 6552 df-1st 7059 df-2nd 7060 df-grpo 26731 df-ablo 26783 df-rngo 32864 |
This theorem is referenced by: rngorn1 32902 |
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