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Mathbox for Jeff Madsen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmnnzd | Structured version Visualization version GIF version |
Description: A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.) |
Ref | Expression |
---|---|
dmnnzd.1 | ⊢ 𝐺 = (1st ‘𝑅) |
dmnnzd.2 | ⊢ 𝐻 = (2nd ‘𝑅) |
dmnnzd.3 | ⊢ 𝑋 = ran 𝐺 |
dmnnzd.4 | ⊢ 𝑍 = (GId‘𝐺) |
Ref | Expression |
---|---|
dmnnzd | ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmnnzd.1 | . . . . . 6 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | dmnnzd.2 | . . . . . 6 ⊢ 𝐻 = (2nd ‘𝑅) | |
3 | dmnnzd.3 | . . . . . 6 ⊢ 𝑋 = ran 𝐺 | |
4 | dmnnzd.4 | . . . . . 6 ⊢ 𝑍 = (GId‘𝐺) | |
5 | eqid 2736 | . . . . . 6 ⊢ (GId‘𝐻) = (GId‘𝐻) | |
6 | 1, 2, 3, 4, 5 | isdmn3 36524 | . . . . 5 ⊢ (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ (GId‘𝐻) ≠ 𝑍 ∧ ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)))) |
7 | 6 | simp3bi 1147 | . . . 4 ⊢ (𝑅 ∈ Dmn → ∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍))) |
8 | oveq1 7363 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎𝐻𝑏) = (𝐴𝐻𝑏)) | |
9 | 8 | eqeq1d 2738 | . . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝑏) = 𝑍)) |
10 | eqeq1 2740 | . . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 = 𝑍 ↔ 𝐴 = 𝑍)) | |
11 | 10 | orbi1d 915 | . . . . . 6 ⊢ (𝑎 = 𝐴 → ((𝑎 = 𝑍 ∨ 𝑏 = 𝑍) ↔ (𝐴 = 𝑍 ∨ 𝑏 = 𝑍))) |
12 | 9, 11 | imbi12d 344 | . . . . 5 ⊢ (𝑎 = 𝐴 → (((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)) ↔ ((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍 ∨ 𝑏 = 𝑍)))) |
13 | oveq2 7364 | . . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝐴𝐻𝑏) = (𝐴𝐻𝐵)) | |
14 | 13 | eqeq1d 2738 | . . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴𝐻𝑏) = 𝑍 ↔ (𝐴𝐻𝐵) = 𝑍)) |
15 | eqeq1 2740 | . . . . . . 7 ⊢ (𝑏 = 𝐵 → (𝑏 = 𝑍 ↔ 𝐵 = 𝑍)) | |
16 | 15 | orbi2d 914 | . . . . . 6 ⊢ (𝑏 = 𝐵 → ((𝐴 = 𝑍 ∨ 𝑏 = 𝑍) ↔ (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))) |
17 | 14, 16 | imbi12d 344 | . . . . 5 ⊢ (𝑏 = 𝐵 → (((𝐴𝐻𝑏) = 𝑍 → (𝐴 = 𝑍 ∨ 𝑏 = 𝑍)) ↔ ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))) |
18 | 12, 17 | rspc2v 3590 | . . . 4 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (∀𝑎 ∈ 𝑋 ∀𝑏 ∈ 𝑋 ((𝑎𝐻𝑏) = 𝑍 → (𝑎 = 𝑍 ∨ 𝑏 = 𝑍)) → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))) |
19 | 7, 18 | syl5com 31 | . . 3 ⊢ (𝑅 ∈ Dmn → ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)))) |
20 | 19 | expd 416 | . 2 ⊢ (𝑅 ∈ Dmn → (𝐴 ∈ 𝑋 → (𝐵 ∈ 𝑋 → ((𝐴𝐻𝐵) = 𝑍 → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍))))) |
21 | 20 | 3imp2 1349 | 1 ⊢ ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ (𝐴𝐻𝐵) = 𝑍)) → (𝐴 = 𝑍 ∨ 𝐵 = 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ran crn 5634 ‘cfv 6496 (class class class)co 7356 1st c1st 7918 2nd c2nd 7919 GIdcgi 29430 CRingOpsccring 36443 Dmncdmn 36497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7920 df-2nd 7921 df-1o 8411 df-en 8883 df-grpo 29433 df-gid 29434 df-ginv 29435 df-ablo 29485 df-ass 36293 df-exid 36295 df-mgmOLD 36299 df-sgrOLD 36311 df-mndo 36317 df-rngo 36345 df-com2 36440 df-crngo 36444 df-idl 36460 df-pridl 36461 df-prrngo 36498 df-dmn 36499 df-igen 36510 |
This theorem is referenced by: dmncan1 36526 |
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