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Mirrors > Home > ILE Home > Th. List > dvdsr02 | 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 | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
2 | 1 | a1i 9 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝐵 = (Base‘𝑅)) |
3 | dvdsr0.d | . . . 4 ⊢ ∥ = (∥r‘𝑅) | |
4 | 3 | a1i 9 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ∥ = (∥r‘𝑅)) |
5 | ringsrg 13229 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ SRing) | |
6 | 5 | adantr 276 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ SRing) |
7 | eqid 2177 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
8 | 7 | a1i 9 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (.r‘𝑅) = (.r‘𝑅)) |
9 | dvdsr0.z | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
10 | 1, 9 | ring0cl 13209 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
11 | 10 | adantr 276 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
12 | 2, 4, 6, 8, 11 | dvdsr2d 13269 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∥ 𝑋 ↔ ∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋)) |
13 | 1, 7, 9 | ringrz 13228 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝑅) 0 ) = 0 ) |
14 | 13 | eqeq1d 2186 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 0 = 𝑋)) |
15 | eqcom 2179 | . . . . . 6 ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 ) | |
16 | 14, 15 | bitrdi 196 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
17 | 16 | rexbidva 2474 | . . . 4 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ ∃𝑥 ∈ 𝐵 𝑋 = 0 )) |
18 | elex2 2755 | . . . . 5 ⊢ ( 0 ∈ 𝐵 → ∃𝑤 𝑤 ∈ 𝐵) | |
19 | r19.9rmv 3516 | . . . . 5 ⊢ (∃𝑤 𝑤 ∈ 𝐵 → (𝑋 = 0 ↔ ∃𝑥 ∈ 𝐵 𝑋 = 0 )) | |
20 | 10, 18, 19 | 3syl 17 | . . . 4 ⊢ (𝑅 ∈ Ring → (𝑋 = 0 ↔ ∃𝑥 ∈ 𝐵 𝑋 = 0 )) |
21 | 17, 20 | bitr4d 191 | . . 3 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
22 | 21 | adantr 276 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
23 | 12, 22 | bitrd 188 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∥ 𝑋 ↔ 𝑋 = 0 )) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∃wrex 2456 class class class wbr 4005 ‘cfv 5218 (class class class)co 5877 Basecbs 12464 .rcmulr 12539 0gc0g 12710 SRingcsrg 13151 Ringcrg 13184 ∥rcdsr 13260 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4120 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-addcom 7913 ax-addass 7915 ax-i2m1 7918 ax-0lt1 7919 ax-0id 7921 ax-rnegex 7922 ax-pre-ltirr 7925 ax-pre-ltadd 7929 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2741 df-sbc 2965 df-csb 3060 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-int 3847 df-iun 3890 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-ltxr 7999 df-inn 8922 df-2 8980 df-3 8981 df-ndx 12467 df-slot 12468 df-base 12470 df-sets 12471 df-plusg 12551 df-mulr 12552 df-0g 12712 df-mgm 12780 df-sgrp 12813 df-mnd 12823 df-grp 12885 df-minusg 12886 df-cmn 13095 df-abl 13096 df-mgp 13136 df-ur 13148 df-srg 13152 df-ring 13186 df-dvdsr 13263 |
This theorem is referenced by: (None) |
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