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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngoid | Structured version Visualization version GIF version |
Description: The multiplication operation of a unital ring has (one or more) identity elements. (Contributed by Steve Rodriguez, 9-Sep-2007.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringi.1 | ⊢ 𝐺 = (1st ‘𝑅) |
ringi.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
ringi.3 | ⊢ 𝑋 = ran 𝐺 |
Ref | Expression |
---|---|
rngoid | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringi.1 | . . . . 5 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | ringi.2 | . . . . 5 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | ringi.3 | . . . . 5 ⊢ 𝑋 = ran 𝐺 | |
4 | 1, 2, 3 | rngoi 35179 | . . . 4 ⊢ (𝑅 ∈ RingOps → ((𝐺 ∈ AbelOp ∧ 𝐻:(𝑋 × 𝑋)⟶𝑋) ∧ (∀𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (((𝑢𝐻𝑥)𝐻𝑦) = (𝑢𝐻(𝑥𝐻𝑦)) ∧ (𝑢𝐻(𝑥𝐺𝑦)) = ((𝑢𝐻𝑥)𝐺(𝑢𝐻𝑦)) ∧ ((𝑢𝐺𝑥)𝐻𝑦) = ((𝑢𝐻𝑦)𝐺(𝑥𝐻𝑦))) ∧ ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)))) |
5 | 4 | simprrd 772 | . . 3 ⊢ (𝑅 ∈ RingOps → ∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) |
6 | r19.12 3326 | . . 3 ⊢ (∃𝑢 ∈ 𝑋 ∀𝑥 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝑅 ∈ RingOps → ∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥)) |
8 | oveq2 7166 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑢𝐻𝑥) = (𝑢𝐻𝐴)) | |
9 | id 22 | . . . . . 6 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
10 | 8, 9 | eqeq12d 2839 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑢𝐻𝑥) = 𝑥 ↔ (𝑢𝐻𝐴) = 𝐴)) |
11 | oveq1 7165 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝐻𝑢) = (𝐴𝐻𝑢)) | |
12 | 11, 9 | eqeq12d 2839 | . . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑥𝐻𝑢) = 𝑥 ↔ (𝐴𝐻𝑢) = 𝐴)) |
13 | 10, 12 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝐴 → (((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))) |
14 | 13 | rexbidv 3299 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ↔ ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴))) |
15 | 14 | rspccva 3624 | . 2 ⊢ ((∀𝑥 ∈ 𝑋 ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝑥) = 𝑥 ∧ (𝑥𝐻𝑢) = 𝑥) ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴)) |
16 | 7, 15 | sylan 582 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑢 ∈ 𝑋 ((𝑢𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑢) = 𝐴)) |
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-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-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-fv 6365 df-ov 7161 df-1st 7691 df-2nd 7692 df-rngo 35175 |
This theorem is referenced by: rngo2 35187 |
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