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 6678 | . . . . . 6 ⊢ 𝑍 ∈ V |
3 | 2 | snid 4594 | . . . . 5 ⊢ 𝑍 ∈ {𝑍} |
4 | eleq1 2900 | . . . . 5 ⊢ (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍})) | |
5 | 3, 4 | mpbii 235 | . . . 4 ⊢ (𝑍 = 𝑈 → 𝑈 ∈ {𝑍}) |
6 | 0ring.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
7 | 6, 1 | 0idl 35297 | . . . . 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 35298 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋)) |
12 | 7, 11 | mpdan 685 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋)) |
13 | 5, 12 | syl5ib 246 | . . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋)) |
14 | eqcom 2828 | . . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍}) | |
15 | 13, 14 | syl6ib 253 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍})) |
16 | 6 | rneqi 5801 | . . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
17 | 9, 16 | eqtri 2844 | . . . 4 ⊢ 𝑋 = ran (1st ‘𝑅) |
18 | 17, 8, 10 | rngo1cl 35211 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
19 | eleq2 2901 | . . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍})) | |
20 | elsni 4577 | . . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍) | |
21 | 20 | eqcomd 2827 | . . . 4 ⊢ (𝑈 ∈ {𝑍} → 𝑍 = 𝑈) |
22 | 19, 21 | syl6bi 255 | . . 3 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑍 = 𝑈)) |
23 | 18, 22 | syl5com 31 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈)) |
24 | 15, 23 | impbid 214 | 1 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 {csn 4560 ran crn 5550 ‘cfv 6349 1st c1st 7681 2nd c2nd 7682 GIdcgi 28261 RingOpscrngo 35166 Idlcidl 35279 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-1st 7683 df-2nd 7684 df-grpo 28264 df-gid 28265 df-ginv 28266 df-ablo 28316 df-ass 35115 df-exid 35117 df-mgmOLD 35121 df-sgrOLD 35133 df-mndo 35139 df-rngo 35167 df-idl 35282 |
This theorem is referenced by: smprngopr 35324 isfldidl2 35341 |
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