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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 6919 | . . . . . 6 ⊢ 𝑍 ∈ V | 
| 3 | 2 | snid 4661 | . . . . 5 ⊢ 𝑍 ∈ {𝑍} | 
| 4 | eleq1 2828 | . . . . 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 38033 | . . . . 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 38034 | . . . . 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 2743 | . . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍}) | |
| 15 | 13, 14 | imbitrdi 251 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍})) | 
| 16 | 6 | rneqi 5947 | . . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅) | 
| 17 | 9, 16 | eqtri 2764 | . . . 4 ⊢ 𝑋 = ran (1st ‘𝑅) | 
| 18 | 17, 8, 10 | rngo1cl 37947 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) | 
| 19 | eleq2 2829 | . . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍})) | |
| 20 | elsni 4642 | . . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍) | |
| 21 | 20 | eqcomd 2742 | . . . 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 1539 ∈ wcel 2107 {csn 4625 ran crn 5685 ‘cfv 6560 1st c1st 8013 2nd c2nd 8014 GIdcgi 30510 RingOpscrngo 37902 Idlcidl 38015 | 
| 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-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 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-rmo 3379 df-reu 3380 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-pw 4601 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-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-1st 8015 df-2nd 8016 df-grpo 30513 df-gid 30514 df-ginv 30515 df-ablo 30565 df-ass 37851 df-exid 37853 df-mgmOLD 37857 df-sgrOLD 37869 df-mndo 37875 df-rngo 37903 df-idl 38018 | 
| This theorem is referenced by: smprngopr 38060 isfldidl2 38077 | 
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