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Mirrors > Home > MPE Home > Th. List > dvdsr02 | Structured version Visualization version GIF version |
Description: Only zero is divisible by zero. (Contributed by Stefan O'Rear, 29-Mar-2015.) |
Ref | Expression |
---|---|
dvdsr0.b | ⊢ 𝐵 = (Base‘𝑅) |
dvdsr0.d | ⊢ ∥ = (∥r‘𝑅) |
dvdsr0.z | ⊢ 0 = (0g‘𝑅) |
Ref | Expression |
---|---|
dvdsr02 | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∥ 𝑋 ↔ 𝑋 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dvdsr0.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
2 | dvdsr0.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
3 | 1, 2 | ring0cl 18923 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
4 | 3 | adantr 474 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
5 | dvdsr0.d | . . . 4 ⊢ ∥ = (∥r‘𝑅) | |
6 | eqid 2825 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | 1, 5, 6 | dvdsr2 19001 | . . 3 ⊢ ( 0 ∈ 𝐵 → ( 0 ∥ 𝑋 ↔ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋)) |
8 | 4, 7 | syl 17 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∥ 𝑋 ↔ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋)) |
9 | 1, 6, 2 | ringrz 18942 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝑅) 0 ) = 0 ) |
10 | 9 | eqeq1d 2827 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 0 = 𝑋)) |
11 | eqcom 2832 | . . . . . 6 ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 ) | |
12 | 10, 11 | syl6bb 279 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
13 | 12 | rexbidva 3259 | . . . 4 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ ∃𝑥 ∈ 𝐵 𝑋 = 0 )) |
14 | ringgrp 18906 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
15 | 1 | grpbn0 17805 | . . . . 5 ⊢ (𝑅 ∈ Grp → 𝐵 ≠ ∅) |
16 | r19.9rzv 4287 | . . . . 5 ⊢ (𝐵 ≠ ∅ → (𝑋 = 0 ↔ ∃𝑥 ∈ 𝐵 𝑋 = 0 )) | |
17 | 14, 15, 16 | 3syl 18 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑋 = 0 ↔ ∃𝑥 ∈ 𝐵 𝑋 = 0 )) |
18 | 13, 17 | bitr4d 274 | . . 3 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
19 | 18 | adantr 474 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
20 | 8, 19 | bitrd 271 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∥ 𝑋 ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2999 ∃wrex 3118 ∅c0 4144 class class class wbr 4873 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 .rcmulr 16306 0gc0g 16453 Grpcgrp 17776 Ringcrg 18901 ∥rcdsr 18992 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-er 8009 df-en 8223 df-dom 8224 df-sdom 8225 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-plusg 16318 df-0g 16455 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-mgp 18844 df-ring 18903 df-dvdsr 18995 |
This theorem is referenced by: (None) |
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