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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 2823 | . . . 4 ⊢ ran 𝐺 = ran 𝐺 | |
4 | 1, 2, 3 | rngosm 35180 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝐻:(ran 𝐺 × ran 𝐺)⟶ran 𝐺) |
5 | 1, 2, 3 | rngoi 35179 | . . . 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 772 | . . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑥 ∈ ran 𝐺∀𝑦 ∈ ran 𝐺((𝑥𝐻𝑦) = 𝑦 ∧ (𝑦𝐻𝑥) = 𝑦)) |
7 | rngmgmbs4 35211 | . . 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 586 | . 2 ⊢ (𝑅 ∈ RingOps → ran 𝐻 = ran 𝐺) |
9 | 8 | eqcomd 2829 | 1 ⊢ (𝑅 ∈ RingOps → ran 𝐺 = ran 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3140 ∃wrex 3141 × cxp 5555 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1st c1st 7689 2nd c2nd 7690 AbelOpcablo 28323 RingOpscrngo 35174 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fo 6363 df-fv 6365 df-ov 7161 df-1st 7691 df-2nd 7692 df-rngo 35175 |
This theorem is referenced by: rngoidmlem 35216 rngo1cl 35219 isdrngo2 35238 |
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