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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngo2 | Structured version Visualization version GIF version |
Description: A ring element plus itself is two times the element. (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 |
---|---|
rngo2 | ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringi.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | ringi.2 | . . 3 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | ringi.3 | . . 3 ⊢ 𝑋 = ran 𝐺 | |
4 | 1, 2, 3 | rngoid 36765 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 ((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴)) |
5 | oveq12 7417 | . . . . . . 7 ⊢ (((𝑥𝐻𝐴) = 𝐴 ∧ (𝑥𝐻𝐴) = 𝐴) → ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)) = (𝐴𝐺𝐴)) | |
6 | 5 | anidms 567 | . . . . . 6 ⊢ ((𝑥𝐻𝐴) = 𝐴 → ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)) = (𝐴𝐺𝐴)) |
7 | 6 | eqcomd 2738 | . . . . 5 ⊢ ((𝑥𝐻𝐴) = 𝐴 → (𝐴𝐺𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴))) |
8 | simpll 765 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑅 ∈ RingOps) | |
9 | simpr 485 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋) | |
10 | simplr 767 | . . . . . . 7 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ 𝑋) | |
11 | 1, 2, 3 | rngodir 36768 | . . . . . . 7 ⊢ ((𝑅 ∈ RingOps ∧ (𝑥 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑥𝐺𝑥)𝐻𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴))) |
12 | 8, 9, 9, 10, 11 | syl13anc 1372 | . . . . . 6 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐺𝑥)𝐻𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴))) |
13 | 12 | eqeq2d 2743 | . . . . 5 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴) ↔ (𝐴𝐺𝐴) = ((𝑥𝐻𝐴)𝐺(𝑥𝐻𝐴)))) |
14 | 7, 13 | imbitrrid 245 | . . . 4 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → ((𝑥𝐻𝐴) = 𝐴 → (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))) |
15 | 14 | adantrd 492 | . . 3 ⊢ (((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) ∧ 𝑥 ∈ 𝑋) → (((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴) → (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))) |
16 | 15 | reximdva 3168 | . 2 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → (∃𝑥 ∈ 𝑋 ((𝑥𝐻𝐴) = 𝐴 ∧ (𝐴𝐻𝑥) = 𝐴) → ∃𝑥 ∈ 𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴))) |
17 | 4, 16 | mpd 15 | 1 ⊢ ((𝑅 ∈ RingOps ∧ 𝐴 ∈ 𝑋) → ∃𝑥 ∈ 𝑋 (𝐴𝐺𝐴) = ((𝑥𝐺𝑥)𝐻𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ran crn 5677 ‘cfv 6543 (class class class)co 7408 1st c1st 7972 2nd c2nd 7973 RingOpscrngo 36757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7411 df-1st 7974 df-2nd 7975 df-rngo 36758 |
This theorem is referenced by: (None) |
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