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Mirrors > Home > MPE Home > Th. List > drngmuleq0 | Structured version Visualization version GIF version |
Description: An element is zero iff its product with a nonzero element is zero. (Contributed by NM, 8-Oct-2014.) |
Ref | Expression |
---|---|
drngmuleq0.b | ⊢ 𝐵 = (Base‘𝑅) |
drngmuleq0.o | ⊢ 0 = (0g‘𝑅) |
drngmuleq0.t | ⊢ · = (.r‘𝑅) |
drngmuleq0.r | ⊢ (𝜑 → 𝑅 ∈ DivRing) |
drngmuleq0.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
drngmuleq0.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
drngmuleq0.e | ⊢ (𝜑 → 𝑌 ≠ 0 ) |
Ref | Expression |
---|---|
drngmuleq0 | ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑋 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | drngmuleq0.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
2 | drngmuleq0.o | . . 3 ⊢ 0 = (0g‘𝑅) | |
3 | drngmuleq0.t | . . 3 ⊢ · = (.r‘𝑅) | |
4 | drngmuleq0.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ DivRing) | |
5 | drngmuleq0.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | drngmuleq0.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
7 | 1, 2, 3, 4, 5, 6 | drngmul0or 20777 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ (𝑋 = 0 ∨ 𝑌 = 0 ))) |
8 | drngmuleq0.e | . . 3 ⊢ (𝜑 → 𝑌 ≠ 0 ) | |
9 | df-ne 2939 | . . . 4 ⊢ (𝑌 ≠ 0 ↔ ¬ 𝑌 = 0 ) | |
10 | orel2 890 | . . . . 5 ⊢ (¬ 𝑌 = 0 → ((𝑋 = 0 ∨ 𝑌 = 0 ) → 𝑋 = 0 )) | |
11 | orc 867 | . . . . 5 ⊢ (𝑋 = 0 → (𝑋 = 0 ∨ 𝑌 = 0 )) | |
12 | 10, 11 | impbid1 225 | . . . 4 ⊢ (¬ 𝑌 = 0 → ((𝑋 = 0 ∨ 𝑌 = 0 ) ↔ 𝑋 = 0 )) |
13 | 9, 12 | sylbi 217 | . . 3 ⊢ (𝑌 ≠ 0 → ((𝑋 = 0 ∨ 𝑌 = 0 ) ↔ 𝑋 = 0 )) |
14 | 8, 13 | syl 17 | . 2 ⊢ (𝜑 → ((𝑋 = 0 ∨ 𝑌 = 0 ) ↔ 𝑋 = 0 )) |
15 | 7, 14 | bitrd 279 | 1 ⊢ (𝜑 → ((𝑋 · 𝑌) = 0 ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 .rcmulr 17299 0gc0g 17486 DivRingcdr 20746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 |
This theorem is referenced by: lkrsc 39079 |
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