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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > 0rngo | Structured version Visualization version GIF version |
Description: In a ring, 0 = 1 iff the ring contains only 0. (Contributed by Jeff Madsen, 6-Jan-2011.) |
Ref | Expression |
---|---|
0ring.1 | ⊢ 𝐺 = (1st ‘𝑅) |
0ring.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
0ring.3 | ⊢ 𝑋 = ran 𝐺 |
0ring.4 | ⊢ 𝑍 = (GId‘𝐺) |
0ring.5 | ⊢ 𝑈 = (GId‘𝐻) |
Ref | Expression |
---|---|
0rngo | ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ring.4 | . . . . . . 7 ⊢ 𝑍 = (GId‘𝐺) | |
2 | 1 | fvexi 6921 | . . . . . 6 ⊢ 𝑍 ∈ V |
3 | 2 | snid 4667 | . . . . 5 ⊢ 𝑍 ∈ {𝑍} |
4 | eleq1 2827 | . . . . 5 ⊢ (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍})) | |
5 | 3, 4 | mpbii 233 | . . . 4 ⊢ (𝑍 = 𝑈 → 𝑈 ∈ {𝑍}) |
6 | 0ring.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
7 | 6, 1 | 0idl 38012 | . . . . 5 ⊢ (𝑅 ∈ RingOps → {𝑍} ∈ (Idl‘𝑅)) |
8 | 0ring.2 | . . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅) | |
9 | 0ring.3 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
10 | 0ring.5 | . . . . . 6 ⊢ 𝑈 = (GId‘𝐻) | |
11 | 6, 8, 9, 10 | 1idl 38013 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋)) |
12 | 7, 11 | mpdan 687 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋)) |
13 | 5, 12 | imbitrid 244 | . . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋)) |
14 | eqcom 2742 | . . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍}) | |
15 | 13, 14 | imbitrdi 251 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍})) |
16 | 6 | rneqi 5951 | . . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
17 | 9, 16 | eqtri 2763 | . . . 4 ⊢ 𝑋 = ran (1st ‘𝑅) |
18 | 17, 8, 10 | rngo1cl 37926 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
19 | eleq2 2828 | . . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍})) | |
20 | elsni 4648 | . . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍) | |
21 | 20 | eqcomd 2741 | . . . 4 ⊢ (𝑈 ∈ {𝑍} → 𝑍 = 𝑈) |
22 | 19, 21 | biimtrdi 253 | . . 3 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑍 = 𝑈)) |
23 | 18, 22 | syl5com 31 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈)) |
24 | 15, 23 | impbid 212 | 1 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 {csn 4631 ran crn 5690 ‘cfv 6563 1st c1st 8011 2nd c2nd 8012 GIdcgi 30519 RingOpscrngo 37881 Idlcidl 37994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-1st 8013 df-2nd 8014 df-grpo 30522 df-gid 30523 df-ginv 30524 df-ablo 30574 df-ass 37830 df-exid 37832 df-mgmOLD 37836 df-sgrOLD 37848 df-mndo 37854 df-rngo 37882 df-idl 37997 |
This theorem is referenced by: smprngopr 38039 isfldidl2 38056 |
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