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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rngorn1eq | Structured version Visualization version GIF version | ||
| Description: In a unital ring the range of the addition equals the range 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 | 
|---|---|
| rngorn1eq | ⊢ (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rnplrnml0.2 | . . . 4 ⊢ 𝐺 = (1st ‘𝑅) | |
| 2 | rnplrnml0.1 | . . . 4 ⊢ 𝐻 = (2nd ‘𝑅) | |
| 3 | eqid 2736 | . . . 4 ⊢ ran 𝐺 = ran 𝐺 | |
| 4 | 1, 2, 3 | rngosm 37908 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) | 
| 5 | 1, 2, 3 | rngoi 37907 | . . . 4 ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) ∧ (∀𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺∀𝑧 ∈ ran 𝐺(((𝑥𝐻𝑦)𝐻𝑧) = (𝑥𝐻(𝑦𝐻𝑧)) ∧ (𝑥𝐻(𝑦𝐺𝑧)) = ((𝑥𝐻𝑦)𝐺(𝑥𝐻𝑧)) ∧ ((𝑥𝐺𝑦)𝐻𝑧) = ((𝑥𝐻𝑧)𝐺(𝑦𝐻𝑧))) ∧ ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)))) | 
| 6 | 5 | simprrd 773 | . . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) | 
| 7 | rngmgmbs4 37939 | . . 3 ⊢ ((𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺 ∧ ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) → ran 𝐻 = ran 𝐺) | |
| 8 | 4, 6, 7 | syl2anc 584 | . 2 ⊢ (𝑅 ∈ RingOps → ran 𝐻 = ran 𝐺) | 
| 9 | 8 | eqcomd 2742 | 1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ∀wral 3060 ∃wrex 3069 × cxp 5682 ran crn 5685 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 1st c1st 8013 2nd c2nd 8014 AbelOpcablo 30564 RingOpscrngo 37902 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fo 6566 df-fv 6568 df-ov 7435 df-1st 8015 df-2nd 8016 df-rngo 37903 | 
| This theorem is referenced by: rngoidmlem 37944 rngo1cl 37947 isdrngo2 37966 | 
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