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Mirrors > Home > MPE Home > Th. List > domnmuln0 | Structured version Visualization version GIF version |
Description: In a domain, a product of nonzero elements is nonzero. (Contributed by Mario Carneiro, 6-May-2015.) |
Ref | Expression |
---|---|
domneq0.b | ⊢ 𝐵 = (Base‘𝑅) |
domneq0.t | ⊢ · = (.r‘𝑅) |
domneq0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
domnmuln0 | ⊢ ((𝑅 ∈ Domn ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) → (𝑋 · 𝑌) ≠ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | an4 654 | . . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) ↔ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ))) | |
2 | neanior 3025 | . . . . . 6 ⊢ ((𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ↔ ¬ (𝑋 = 0 ∨ 𝑌 = 0 )) | |
3 | domneq0.b | . . . . . . . . 9 ⊢ 𝐵 = (Base‘𝑅) | |
4 | domneq0.t | . . . . . . . . 9 ⊢ · = (.r‘𝑅) | |
5 | domneq0.z | . . . . . . . . 9 ⊢ 0 = (0g‘𝑅) | |
6 | 3, 4, 5 | domneq0 20686 | . . . . . . . 8 ⊢ ((𝑅 ∈ Domn ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) |
7 | 6 | 3expb 1117 | . . . . . . 7 ⊢ ((𝑅 ∈ Domn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) |
8 | 7 | necon3abid 2967 | . . . . . 6 ⊢ ((𝑅 ∈ Domn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 · 𝑌) ≠ 0 ↔ ¬ (𝑋 = 0 ∨ 𝑌 = 0 ))) |
9 | 2, 8 | bitr4id 289 | . . . . 5 ⊢ ((𝑅 ∈ Domn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) ↔ (𝑋 · 𝑌) ≠ 0 )) |
10 | 9 | biimpd 228 | . . . 4 ⊢ ((𝑅 ∈ Domn ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 ≠ 0 ∧ 𝑌 ≠ 0 ) → (𝑋 · 𝑌) ≠ 0 )) |
11 | 10 | expimpd 452 | . . 3 ⊢ (𝑅 ∈ Domn → (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≠ 0 ∧ 𝑌 ≠ 0 )) → (𝑋 · 𝑌) ≠ 0 )) |
12 | 1, 11 | biimtrid 241 | . 2 ⊢ (𝑅 ∈ Domn → (((𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) → (𝑋 · 𝑌) ≠ 0 )) |
13 | 12 | 3impib 1113 | 1 ⊢ ((𝑅 ∈ Domn ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≠ 0 ) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≠ 0 )) → (𝑋 · 𝑌) ≠ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 .rcmulr 17267 0gc0g 17454 Domncdomn 20670 |
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 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-2nd 8004 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-er 8734 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-plusg 17279 df-0g 17456 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-grp 18931 df-minusg 18932 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-nzr 20495 df-domn 20673 |
This theorem is referenced by: isdomn4 20694 drngmcl 20728 abvtrivg 20812 domnprodn0 33130 idomnnzpownz 41830 idomnnzgmulnz 41831 aks6d1c5lem2 41836 deg1mhm 42865 domnmsuppn0 47748 |
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