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Mirrors > Home > MPE Home > Th. List > dvr1 | Structured version Visualization version GIF version |
Description: A cancellation law for division. (div1 11130 analog.) (Contributed by Mario Carneiro, 18-Jun-2015.) |
Ref | Expression |
---|---|
dvr1.b | ⊢ 𝐵 = (Base‘𝑅) |
dvr1.d | ⊢ / = (/r‘𝑅) |
dvr1.o | ⊢ 1 = (1r‘𝑅) |
Ref | Expression |
---|---|
dvr1 | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 / 1 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑋 ∈ 𝐵 → 𝑋 ∈ 𝐵) | |
2 | eqid 2779 | . . . 4 ⊢ (Unit‘𝑅) = (Unit‘𝑅) | |
3 | dvr1.o | . . . 4 ⊢ 1 = (1r‘𝑅) | |
4 | 2, 3 | 1unit 19131 | . . 3 ⊢ (𝑅 ∈ Ring → 1 ∈ (Unit‘𝑅)) |
5 | dvr1.b | . . . 4 ⊢ 𝐵 = (Base‘𝑅) | |
6 | eqid 2779 | . . . 4 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
7 | eqid 2779 | . . . 4 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
8 | dvr1.d | . . . 4 ⊢ / = (/r‘𝑅) | |
9 | 5, 6, 2, 7, 8 | dvrval 19158 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 1 ∈ (Unit‘𝑅)) → (𝑋 / 1 ) = (𝑋(.r‘𝑅)((invr‘𝑅)‘ 1 ))) |
10 | 1, 4, 9 | syl2anr 587 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 / 1 ) = (𝑋(.r‘𝑅)((invr‘𝑅)‘ 1 ))) |
11 | 7, 3 | 1rinv 19152 | . . . 4 ⊢ (𝑅 ∈ Ring → ((invr‘𝑅)‘ 1 ) = 1 ) |
12 | 11 | adantr 473 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → ((invr‘𝑅)‘ 1 ) = 1 ) |
13 | 12 | oveq2d 6992 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(.r‘𝑅)((invr‘𝑅)‘ 1 )) = (𝑋(.r‘𝑅) 1 )) |
14 | 5, 6, 3 | ringridm 19045 | . 2 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋(.r‘𝑅) 1 ) = 𝑋) |
15 | 10, 13, 14 | 3eqtrd 2819 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵) → (𝑋 / 1 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ‘cfv 6188 (class class class)co 6976 Basecbs 16339 .rcmulr 16422 1rcur 18974 Ringcrg 19020 Unitcui 19112 invrcinvr 19144 /rcdvr 19155 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2751 ax-rep 5049 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-1cn 10393 ax-icn 10394 ax-addcl 10395 ax-addrcl 10396 ax-mulcl 10397 ax-mulrcl 10398 ax-mulcom 10399 ax-addass 10400 ax-mulass 10401 ax-distr 10402 ax-i2m1 10403 ax-1ne0 10404 ax-1rid 10405 ax-rnegex 10406 ax-rrecex 10407 ax-cnre 10408 ax-pre-lttri 10409 ax-pre-lttrn 10410 ax-pre-ltadd 10411 ax-pre-mulgt0 10412 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2760 df-cleq 2772 df-clel 2847 df-nfc 2919 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3418 df-sbc 3683 df-csb 3788 df-dif 3833 df-un 3835 df-in 3837 df-ss 3844 df-pss 3846 df-nul 4180 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-tp 4446 df-op 4448 df-uni 4713 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-tr 5031 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-fr 5366 df-we 5368 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-pred 5986 df-ord 6032 df-on 6033 df-lim 6034 df-suc 6035 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-riota 6937 df-ov 6979 df-oprab 6980 df-mpo 6981 df-om 7397 df-1st 7501 df-2nd 7502 df-tpos 7695 df-wrecs 7750 df-recs 7812 df-rdg 7850 df-er 8089 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-sub 10672 df-neg 10673 df-nn 11440 df-2 11503 df-3 11504 df-ndx 16342 df-slot 16343 df-base 16345 df-sets 16346 df-ress 16347 df-plusg 16434 df-mulr 16435 df-0g 16571 df-mgm 17710 df-sgrp 17752 df-mnd 17763 df-grp 17894 df-minusg 17895 df-mgp 18963 df-ur 18975 df-ring 19022 df-oppr 19096 df-dvdsr 19114 df-unit 19115 df-invr 19145 df-dvr 19156 |
This theorem is referenced by: qqh0 30866 qqh1 30867 |
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