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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 6857 | . . . . . 6 ⊢ 𝑍 ∈ V |
3 | 2 | snid 4623 | . . . . 5 ⊢ 𝑍 ∈ {𝑍} |
4 | eleq1 2826 | . . . . 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 36487 | . . . . 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 36488 | . . . . 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 2744 | . . 3 ⊢ ({𝑍} = 𝑋 ↔ 𝑋 = {𝑍}) | |
15 | 13, 14 | syl6ib 251 | . 2 ⊢ (𝑅 ∈ RingOps → (𝑍 = 𝑈 → 𝑋 = {𝑍})) |
16 | 6 | rneqi 5893 | . . . . 5 ⊢ ran 𝐺 = ran (1st ‘𝑅) |
17 | 9, 16 | eqtri 2765 | . . . 4 ⊢ 𝑋 = ran (1st ‘𝑅) |
18 | 17, 8, 10 | rngo1cl 36401 | . . 3 ⊢ (𝑅 ∈ RingOps → 𝑈 ∈ 𝑋) |
19 | eleq2 2827 | . . . 4 ⊢ (𝑋 = {𝑍} → (𝑈 ∈ 𝑋 ↔ 𝑈 ∈ {𝑍})) | |
20 | elsni 4604 | . . . . 5 ⊢ (𝑈 ∈ {𝑍} → 𝑈 = 𝑍) | |
21 | 20 | eqcomd 2743 | . . . 4 ⊢ (𝑈 ∈ {𝑍} → 𝑍 = 𝑈) |
22 | 19, 21 | syl6bi 253 | . . 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 1542 ∈ wcel 2107 {csn 4587 ran crn 5635 ‘cfv 6497 1st c1st 7920 2nd c2nd 7921 GIdcgi 29435 RingOpscrngo 36356 Idlcidl 36469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-1st 7922 df-2nd 7923 df-grpo 29438 df-gid 29439 df-ginv 29440 df-ablo 29490 df-ass 36305 df-exid 36307 df-mgmOLD 36311 df-sgrOLD 36323 df-mndo 36329 df-rngo 36357 df-idl 36472 |
This theorem is referenced by: smprngopr 36514 isfldidl2 36531 |
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