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 6709 | . . . . . 6 ⊢ 𝑍 ∈ V |
3 | 2 | snid 4563 | . . . . 5 ⊢ 𝑍 ∈ {𝑍} |
4 | eleq1 2818 | . . . . 5 ⊢ (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍})) | |
5 | 3, 4 | mpbii 236 | . . . 4 ⊢ (𝑍 = 𝑈 → 𝑈 ∈ {𝑍}) |
6 | 0ring.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
7 | 6, 1 | 0idl 35869 | . . . . 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 35870 | . . . . 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 | syl5ib 247 | . . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋)) |
14 | eqcom 2743 | . . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍}) | |
15 | 13, 14 | syl6ib 254 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍})) |
16 | 6 | rneqi 5791 | . . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
17 | 9, 16 | eqtri 2759 | . . . 4 ⊢ 𝑋 = ran (1st ‘𝑅) |
18 | 17, 8, 10 | rngo1cl 35783 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
19 | eleq2 2819 | . . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍})) | |
20 | elsni 4544 | . . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍) | |
21 | 20 | eqcomd 2742 | . . . 4 ⊢ (𝑈 ∈ {𝑍} → 𝑍 = 𝑈) |
22 | 19, 21 | syl6bi 256 | . . 3 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑍 = 𝑈)) |
23 | 18, 22 | syl5com 31 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈)) |
24 | 15, 23 | impbid 215 | 1 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 {csn 4527 ran crn 5537 ‘cfv 6358 1st c1st 7737 2nd c2nd 7738 GIdcgi 28525 RingOpscrngo 35738 Idlcidl 35851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-1st 7739 df-2nd 7740 df-grpo 28528 df-gid 28529 df-ginv 28530 df-ablo 28580 df-ass 35687 df-exid 35689 df-mgmOLD 35693 df-sgrOLD 35705 df-mndo 35711 df-rngo 35739 df-idl 35854 |
This theorem is referenced by: smprngopr 35896 isfldidl2 35913 |
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