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Mirrors > Home > MPE Home > Th. List > abvn0b | Structured version Visualization version GIF version |
Description: Another characterization of domains, hinted at in abvtriv 19615: a nonzero ring is a domain iff it has an absolute value. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
abvn0b.b | ⊢ 𝐴 = (AbsVal‘𝑅) |
Ref | Expression |
---|---|
abvn0b | ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | domnnzr 20071 | . . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | abvn0b.b | . . . . 5 ⊢ 𝐴 = (AbsVal‘𝑅) | |
3 | eqid 2824 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | eqid 2824 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | eqid 2824 | . . . . 5 ⊢ (𝑥 ∈ (Base‘𝑅) ↦ if(𝑥 = (0g‘𝑅), 0, 1)) = (𝑥 ∈ (Base‘𝑅) ↦ if(𝑥 = (0g‘𝑅), 0, 1)) | |
6 | eqid 2824 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | domnring 20072 | . . . . 5 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) | |
8 | 3, 6, 4 | domnmuln0 20074 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ 𝑧 ≠ (0g‘𝑅))) → (𝑦(.r‘𝑅)𝑧) ≠ (0g‘𝑅)) |
9 | 2, 3, 4, 5, 6, 7, 8 | abvtrivd 19614 | . . . 4 ⊢ (𝑅 ∈ Domn → (𝑥 ∈ (Base‘𝑅) ↦ if(𝑥 = (0g‘𝑅), 0, 1)) ∈ 𝐴) |
10 | 9 | ne0d 4304 | . . 3 ⊢ (𝑅 ∈ Domn → 𝐴 ≠ ∅) |
11 | 1, 10 | jca 514 | . 2 ⊢ (𝑅 ∈ Domn → (𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅)) |
12 | n0 4313 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
13 | neanior 3112 | . . . . . . . . 9 ⊢ ((𝑦 ≠ (0g‘𝑅) ∧ 𝑧 ≠ (0g‘𝑅)) ↔ ¬ (𝑦 = (0g‘𝑅) ∨ 𝑧 = (0g‘𝑅))) | |
14 | an4 654 | . . . . . . . . . . 11 ⊢ (((𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑦 ≠ (0g‘𝑅) ∧ 𝑧 ≠ (0g‘𝑅))) ↔ ((𝑦 ∈ (Base‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ 𝑧 ≠ (0g‘𝑅)))) | |
15 | 2, 3, 4, 6 | abvdom 19612 | . . . . . . . . . . . 12 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ 𝑧 ≠ (0g‘𝑅))) → (𝑦(.r‘𝑅)𝑧) ≠ (0g‘𝑅)) |
16 | 15 | 3expib 1118 | . . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → (((𝑦 ∈ (Base‘𝑅) ∧ 𝑦 ≠ (0g‘𝑅)) ∧ (𝑧 ∈ (Base‘𝑅) ∧ 𝑧 ≠ (0g‘𝑅))) → (𝑦(.r‘𝑅)𝑧) ≠ (0g‘𝑅))) |
17 | 14, 16 | syl5bi 244 | . . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (((𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅)) ∧ (𝑦 ≠ (0g‘𝑅) ∧ 𝑧 ≠ (0g‘𝑅))) → (𝑦(.r‘𝑅)𝑧) ≠ (0g‘𝑅))) |
18 | 17 | expdimp 455 | . . . . . . . . 9 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑦 ≠ (0g‘𝑅) ∧ 𝑧 ≠ (0g‘𝑅)) → (𝑦(.r‘𝑅)𝑧) ≠ (0g‘𝑅))) |
19 | 13, 18 | syl5bir 245 | . . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → (¬ (𝑦 = (0g‘𝑅) ∨ 𝑧 = (0g‘𝑅)) → (𝑦(.r‘𝑅)𝑧) ≠ (0g‘𝑅))) |
20 | 19 | necon4bd 3039 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑦(.r‘𝑅)𝑧) = (0g‘𝑅) → (𝑦 = (0g‘𝑅) ∨ 𝑧 = (0g‘𝑅)))) |
21 | 20 | ralrimivva 3194 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑦(.r‘𝑅)𝑧) = (0g‘𝑅) → (𝑦 = (0g‘𝑅) ∨ 𝑧 = (0g‘𝑅)))) |
22 | 21 | exlimiv 1930 | . . . . 5 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑦(.r‘𝑅)𝑧) = (0g‘𝑅) → (𝑦 = (0g‘𝑅) ∨ 𝑧 = (0g‘𝑅)))) |
23 | 12, 22 | sylbi 219 | . . . 4 ⊢ (𝐴 ≠ ∅ → ∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑦(.r‘𝑅)𝑧) = (0g‘𝑅) → (𝑦 = (0g‘𝑅) ∨ 𝑧 = (0g‘𝑅)))) |
24 | 23 | anim2i 618 | . . 3 ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅) → (𝑅 ∈ NzRing ∧ ∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑦(.r‘𝑅)𝑧) = (0g‘𝑅) → (𝑦 = (0g‘𝑅) ∨ 𝑧 = (0g‘𝑅))))) |
25 | 3, 6, 4 | isdomn 20070 | . . 3 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ ∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑦(.r‘𝑅)𝑧) = (0g‘𝑅) → (𝑦 = (0g‘𝑅) ∨ 𝑧 = (0g‘𝑅))))) |
26 | 24, 25 | sylibr 236 | . 2 ⊢ ((𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅) → 𝑅 ∈ Domn) |
27 | 11, 26 | impbii 211 | 1 ⊢ (𝑅 ∈ Domn ↔ (𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1536 ∃wex 1779 ∈ wcel 2113 ≠ wne 3019 ∀wral 3141 ∅c0 4294 ifcif 4470 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 0cc0 10540 1c1 10541 Basecbs 16486 .rcmulr 16569 0gc0g 16716 AbsValcabv 19590 NzRingcnzr 20033 Domncdomn 20056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-ico 12747 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-plusg 16581 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-minusg 18110 df-mgp 19243 df-ring 19302 df-abv 19591 df-nzr 20034 df-domn 20060 |
This theorem is referenced by: nrgdomn 23283 |
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