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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 19318 | . . . 4 ⊢ (𝑅 ∈ Ring → 0 ∈ 𝐵) |
4 | 3 | adantr 483 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → 0 ∈ 𝐵) |
5 | dvdsr0.d | . . . 4 ⊢ ∥ = (∥r‘𝑅) | |
6 | eqid 2821 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | 1, 5, 6 | dvdsr2 19396 | . . 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 19337 | . . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → (𝑥(.r‘𝑅) 0 ) = 0 ) |
10 | 9 | eqeq1d 2823 | . . . . . 6 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 0 = 𝑋)) |
11 | eqcom 2828 | . . . . . 6 ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 ) | |
12 | 10, 11 | syl6bb 289 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑥 ∈ 𝐵) → ((𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
13 | 12 | rexbidva 3296 | . . . 4 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ ∃𝑥 ∈ 𝐵 𝑋 = 0 )) |
14 | ringgrp 19301 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
15 | 1 | grpbn0 18131 | . . . . 5 ⊢ (𝑅 ∈ Grp → 𝐵 ≠ ∅) |
16 | r19.9rzv 4444 | . . . . 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 284 | . . 3 ⊢ (𝑅 ∈ Ring → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
19 | 18 | adantr 483 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(.r‘𝑅) 0 ) = 𝑋 ↔ 𝑋 = 0 )) |
20 | 8, 19 | bitrd 281 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ( 0 ∥ 𝑋 ↔ 𝑋 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∃wrex 3139 ∅c0 4290 class class class wbr 5065 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 .rcmulr 16565 0gc0g 16712 Grpcgrp 18102 Ringcrg 19296 ∥rcdsr 19387 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-plusg 16577 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-mgp 19239 df-ring 19298 df-dvdsr 19390 |
This theorem is referenced by: (None) |
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