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Mirrors > Home > MPE Home > Th. List > dvrass | Structured version Visualization version GIF version |
Description: An associative law for division. (divass 10741 analog.) (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
dvrass.b | ⊢ 𝐵 = (Base‘𝑅) |
dvrass.o | ⊢ 𝑈 = (Unit‘𝑅) |
dvrass.d | ⊢ / = (/r‘𝑅) |
dvrass.t | ⊢ · = (.r‘𝑅) |
Ref | Expression |
---|---|
dvrass | ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 · 𝑌) / 𝑍) = (𝑋 · (𝑌 / 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 472 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → 𝑅 ∈ Ring) | |
2 | simpr1 1087 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → 𝑋 ∈ 𝐵) | |
3 | simpr2 1088 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → 𝑌 ∈ 𝐵) | |
4 | simpr3 1089 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → 𝑍 ∈ 𝑈) | |
5 | dvrass.o | . . . . 5 ⊢ 𝑈 = (Unit‘𝑅) | |
6 | eqid 2651 | . . . . 5 ⊢ (invr‘𝑅) = (invr‘𝑅) | |
7 | dvrass.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
8 | 5, 6, 7 | ringinvcl 18722 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝑈) → ((invr‘𝑅)‘𝑍) ∈ 𝐵) |
9 | 1, 4, 8 | syl2anc 694 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → ((invr‘𝑅)‘𝑍) ∈ 𝐵) |
10 | dvrass.t | . . . 4 ⊢ · = (.r‘𝑅) | |
11 | 7, 10 | ringass 18610 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ((invr‘𝑅)‘𝑍) ∈ 𝐵)) → ((𝑋 · 𝑌) · ((invr‘𝑅)‘𝑍)) = (𝑋 · (𝑌 · ((invr‘𝑅)‘𝑍)))) |
12 | 1, 2, 3, 9, 11 | syl13anc 1368 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 · 𝑌) · ((invr‘𝑅)‘𝑍)) = (𝑋 · (𝑌 · ((invr‘𝑅)‘𝑍)))) |
13 | 7, 10 | ringcl 18607 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
14 | 13 | 3adant3r3 1297 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → (𝑋 · 𝑌) ∈ 𝐵) |
15 | dvrass.d | . . . 4 ⊢ / = (/r‘𝑅) | |
16 | 7, 10, 5, 6, 15 | dvrval 18731 | . . 3 ⊢ (((𝑋 · 𝑌) ∈ 𝐵 ∧ 𝑍 ∈ 𝑈) → ((𝑋 · 𝑌) / 𝑍) = ((𝑋 · 𝑌) · ((invr‘𝑅)‘𝑍))) |
17 | 14, 4, 16 | syl2anc 694 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 · 𝑌) / 𝑍) = ((𝑋 · 𝑌) · ((invr‘𝑅)‘𝑍))) |
18 | 7, 10, 5, 6, 15 | dvrval 18731 | . . . 4 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈) → (𝑌 / 𝑍) = (𝑌 · ((invr‘𝑅)‘𝑍))) |
19 | 3, 4, 18 | syl2anc 694 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → (𝑌 / 𝑍) = (𝑌 · ((invr‘𝑅)‘𝑍))) |
20 | 19 | oveq2d 6706 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → (𝑋 · (𝑌 / 𝑍)) = (𝑋 · (𝑌 · ((invr‘𝑅)‘𝑍)))) |
21 | 12, 17, 20 | 3eqtr4d 2695 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝑈)) → ((𝑋 · 𝑌) / 𝑍) = (𝑋 · (𝑌 / 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 .rcmulr 15989 Ringcrg 18593 Unitcui 18685 invrcinvr 18717 /rcdvr 18728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 |
This theorem is referenced by: dvrcan3 18738 irredrmul 18753 dvrcan5 29921 |
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