Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > abvn0b | Structured version Visualization version GIF version |
Description: Another characterization of domains, hinted at in abvtriv 19612: 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 20068 | . . 3 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ NzRing) | |
2 | abvn0b.b | . . . . 5 ⊢ 𝐴 = (AbsVal‘𝑅) | |
3 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
4 | eqid 2821 | . . . . 5 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
5 | eqid 2821 | . . . . 5 ⊢ (𝑥 ∈ (Base‘𝑅) ↦ if(𝑥 = (0g‘𝑅), 0, 1)) = (𝑥 ∈ (Base‘𝑅) ↦ if(𝑥 = (0g‘𝑅), 0, 1)) | |
6 | eqid 2821 | . . . . 5 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | domnring 20069 | . . . . 5 ⊢ (𝑅 ∈ Domn → 𝑅 ∈ Ring) | |
8 | 3, 6, 4 | domnmuln0 20071 | . . . . 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 19611 | . . . 4 ⊢ (𝑅 ∈ Domn → (𝑥 ∈ (Base‘𝑅) ↦ if(𝑥 = (0g‘𝑅), 0, 1)) ∈ 𝐴) |
10 | 9 | ne0d 4301 | . . 3 ⊢ (𝑅 ∈ Domn → 𝐴 ≠ ∅) |
11 | 1, 10 | jca 514 | . 2 ⊢ (𝑅 ∈ Domn → (𝑅 ∈ NzRing ∧ 𝐴 ≠ ∅)) |
12 | n0 4310 | . . . . 5 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴) | |
13 | neanior 3109 | . . . . . . . . 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 19609 | . . . . . . . . . . . 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 3036 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ (𝑦 ∈ (Base‘𝑅) ∧ 𝑧 ∈ (Base‘𝑅))) → ((𝑦(.r‘𝑅)𝑧) = (0g‘𝑅) → (𝑦 = (0g‘𝑅) ∨ 𝑧 = (0g‘𝑅)))) |
21 | 20 | ralrimivva 3191 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑦(.r‘𝑅)𝑧) = (0g‘𝑅) → (𝑦 = (0g‘𝑅) ∨ 𝑧 = (0g‘𝑅)))) |
22 | 21 | exlimiv 1931 | . . . . 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 20067 | . . 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 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∅c0 4291 ifcif 4467 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 0cc0 10537 1c1 10538 Basecbs 16483 .rcmulr 16566 0gc0g 16713 AbsValcabv 19587 NzRingcnzr 20030 Domncdomn 20053 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ico 12745 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-mgp 19240 df-ring 19299 df-abv 19588 df-nzr 20031 df-domn 20057 |
This theorem is referenced by: nrgdomn 23280 |
Copyright terms: Public domain | W3C validator |