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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 6905 | . . . . . 6 ⊢ 𝑍 ∈ V |
3 | 2 | snid 4660 | . . . . 5 ⊢ 𝑍 ∈ {𝑍} |
4 | eleq1 2816 | . . . . 5 ⊢ (𝑍 = 𝑈 → (𝑍 ∈ {𝑍} ↔ 𝑈 ∈ {𝑍})) | |
5 | 3, 4 | mpbii 232 | . . . 4 ⊢ (𝑍 = 𝑈 → 𝑈 ∈ {𝑍}) |
6 | 0ring.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
7 | 6, 1 | 0idl 37487 | . . . . 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 37488 | . . . . 5 ⊢ ((𝑅 ∈ RingOps ∧ {𝑍} ∈ (Idl‘𝑅)) → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋)) |
12 | 7, 11 | mpdan 686 | . . . 4 ⊢ (𝑅 ∈ RingOps → (𝑈 ∈ {𝑍} ↔ {𝑍} = 𝑋)) |
13 | 5, 12 | imbitrid 243 | . . 3 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → {𝑍} = 𝑋)) |
14 | eqcom 2734 | . . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍}) | |
15 | 13, 14 | imbitrdi 250 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍})) |
16 | 6 | rneqi 5933 | . . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
17 | 9, 16 | eqtri 2755 | . . . 4 ⊢ 𝑋 = ran (1st ‘𝑅) |
18 | 17, 8, 10 | rngo1cl 37401 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
19 | eleq2 2817 | . . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍})) | |
20 | elsni 4641 | . . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍) | |
21 | 20 | eqcomd 2733 | . . . 4 ⊢ (𝑈 ∈ {𝑍} → 𝑍 = 𝑈) |
22 | 19, 21 | biimtrdi 252 | . . 3 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 → 𝑍 = 𝑈)) |
23 | 18, 22 | syl5com 31 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑋 = {𝑍} → 𝑍 = 𝑈)) |
24 | 15, 23 | impbid 211 | 1 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 ↔ 𝑋 = {𝑍})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 {csn 4624 ran crn 5673 ‘cfv 6542 1st c1st 7985 2nd c2nd 7986 GIdcgi 30293 RingOpscrngo 37356 Idlcidl 37469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-1st 7987 df-2nd 7988 df-grpo 30296 df-gid 30297 df-ginv 30298 df-ablo 30348 df-ass 37305 df-exid 37307 df-mgmOLD 37311 df-sgrOLD 37323 df-mndo 37329 df-rngo 37357 df-idl 37472 |
This theorem is referenced by: smprngopr 37514 isfldidl2 37531 |
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