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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 19991 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
4 | 3 | adantr 482 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
5 | dvdsr0.d | . . . 4 ⊢ ∥ = (∥r‘𝑅) | |
6 | eqid 2737 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | 1, 5, 6 | dvdsr2 20077 | . . 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 20013 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝑅) 0 ) = 0 ) |
10 | 9 | eqeq1d 2739 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 0 = 𝑋)) |
11 | eqcom 2744 | . . . . . 6 ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 ) | |
12 | 10, 11 | bitrdi 287 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
13 | 12 | rexbidva 3174 | . . . 4 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ ∃𝑥 ∈ 𝐵 𝑋 = 0 )) |
14 | ringgrp 19970 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
15 | 1 | grpbn0 18780 | . . . . 5 ⊢ (𝑅 ∈ Grp → 𝐵 ≠ ∅) |
16 | r19.9rzv 4458 | . . . . 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 282 | . . 3 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
19 | 18 | adantr 482 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
20 | 8, 19 | bitrd 279 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∥ 𝑋 ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2944 ∃wrex 3074 ∅c0 4283 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 Basecbs 17084 .rcmulr 17135 0gc0g 17322 Grpcgrp 18749 Ringcrg 19965 ∥rcdsr 20068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11108 ax-resscn 11109 ax-1cn 11110 ax-icn 11111 ax-addcl 11112 ax-addrcl 11113 ax-mulcl 11114 ax-mulrcl 11115 ax-mulcom 11116 ax-addass 11117 ax-mulass 11118 ax-distr 11119 ax-i2m1 11120 ax-1ne0 11121 ax-1rid 11122 ax-rnegex 11123 ax-rrecex 11124 ax-cnre 11125 ax-pre-lttri 11126 ax-pre-lttrn 11127 ax-pre-ltadd 11128 ax-pre-mulgt0 11129 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3354 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8649 df-en 8885 df-dom 8886 df-sdom 8887 df-pnf 11192 df-mnf 11193 df-xr 11194 df-ltxr 11195 df-le 11196 df-sub 11388 df-neg 11389 df-nn 12155 df-2 12217 df-sets 17037 df-slot 17055 df-ndx 17067 df-base 17085 df-plusg 17147 df-0g 17324 df-mgm 18498 df-sgrp 18547 df-mnd 18558 df-grp 18752 df-mgp 19898 df-ring 19967 df-dvdsr 20071 |
This theorem is referenced by: (None) |
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