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Mirrors > Home > MPE Home > Th. List > ringsubdi | Structured version Visualization version GIF version |
Description: Ring multiplication distributes over subtraction. (subdi 10501 analog.) (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 2-Jul-2014.) |
Ref | Expression |
---|---|
ringsubdi.b | ⊢ 𝐵 = (Base‘𝑅) |
ringsubdi.t | ⊢ · = (.r‘𝑅) |
ringsubdi.m | ⊢ − = (-g‘𝑅) |
ringsubdi.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
ringsubdi.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ringsubdi.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ringsubdi.z | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
Ref | Expression |
---|---|
ringsubdi | ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringsubdi.r | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
2 | ringsubdi.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | ringsubdi.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | ringgrp 18598 | . . . . . 6 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
5 | 1, 4 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Grp) |
6 | ringsubdi.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
7 | ringsubdi.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
8 | eqid 2651 | . . . . . 6 ⊢ (invg‘𝑅) = (invg‘𝑅) | |
9 | 7, 8 | grpinvcl 17514 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((invg‘𝑅)‘𝑍) ∈ 𝐵) |
10 | 5, 6, 9 | syl2anc 694 | . . . 4 ⊢ (𝜑 → ((invg‘𝑅)‘𝑍) ∈ 𝐵) |
11 | eqid 2651 | . . . . 5 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
12 | ringsubdi.t | . . . . 5 ⊢ · = (.r‘𝑅) | |
13 | 7, 11, 12 | ringdi 18612 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ ((invg‘𝑅)‘𝑍) ∈ 𝐵)) → (𝑋 · (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) = ((𝑋 · 𝑌)(+g‘𝑅)(𝑋 · ((invg‘𝑅)‘𝑍)))) |
14 | 1, 2, 3, 10, 13 | syl13anc 1368 | . . 3 ⊢ (𝜑 → (𝑋 · (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) = ((𝑋 · 𝑌)(+g‘𝑅)(𝑋 · ((invg‘𝑅)‘𝑍)))) |
15 | 7, 12, 8, 1, 2, 6 | ringmneg2 18643 | . . . 4 ⊢ (𝜑 → (𝑋 · ((invg‘𝑅)‘𝑍)) = ((invg‘𝑅)‘(𝑋 · 𝑍))) |
16 | 15 | oveq2d 6706 | . . 3 ⊢ (𝜑 → ((𝑋 · 𝑌)(+g‘𝑅)(𝑋 · ((invg‘𝑅)‘𝑍))) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
17 | 14, 16 | eqtrd 2685 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
18 | ringsubdi.m | . . . . 5 ⊢ − = (-g‘𝑅) | |
19 | 7, 11, 8, 18 | grpsubval 17512 | . . . 4 ⊢ ((𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 − 𝑍) = (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) |
20 | 3, 6, 19 | syl2anc 694 | . . 3 ⊢ (𝜑 → (𝑌 − 𝑍) = (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍))) |
21 | 20 | oveq2d 6706 | . 2 ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = (𝑋 · (𝑌(+g‘𝑅)((invg‘𝑅)‘𝑍)))) |
22 | 7, 12 | ringcl 18607 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 · 𝑌) ∈ 𝐵) |
23 | 1, 2, 3, 22 | syl3anc 1366 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑌) ∈ 𝐵) |
24 | 7, 12 | ringcl 18607 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 · 𝑍) ∈ 𝐵) |
25 | 1, 2, 6, 24 | syl3anc 1366 | . . 3 ⊢ (𝜑 → (𝑋 · 𝑍) ∈ 𝐵) |
26 | 7, 11, 8, 18 | grpsubval 17512 | . . 3 ⊢ (((𝑋 · 𝑌) ∈ 𝐵 ∧ (𝑋 · 𝑍) ∈ 𝐵) → ((𝑋 · 𝑌) − (𝑋 · 𝑍)) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
27 | 23, 25, 26 | syl2anc 694 | . 2 ⊢ (𝜑 → ((𝑋 · 𝑌) − (𝑋 · 𝑍)) = ((𝑋 · 𝑌)(+g‘𝑅)((invg‘𝑅)‘(𝑋 · 𝑍)))) |
28 | 17, 21, 27 | 3eqtr4d 2695 | 1 ⊢ (𝜑 → (𝑋 · (𝑌 − 𝑍)) = ((𝑋 · 𝑌) − (𝑋 · 𝑍))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 .rcmulr 15989 Grpcgrp 17469 invgcminusg 17470 -gcsg 17471 Ringcrg 18593 |
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-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-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-plusg 16001 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-sbg 17474 df-mgp 18536 df-ur 18548 df-ring 18595 |
This theorem is referenced by: 2idlcpbl 19282 mdetuni0 20475 chfacfpmmulgsum2 20718 nrgdsdi 22516 nrginvrcnlem 22542 ply1divmo 23940 ornglmulle 29933 |
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