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| Mirrors > Home > MPE Home > Th. List > mulgass3 | Structured version Visualization version GIF version | ||
| Description: An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.) | 
| Ref | Expression | 
|---|---|
| mulgass3.b | ⊢ 𝐵 = (Base‘𝑅) | 
| mulgass3.m | ⊢ · = (.g‘𝑅) | 
| mulgass3.t | ⊢ × = (.r‘𝑅) | 
| Ref | Expression | 
|---|---|
| mulgass3 | ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqid 2737 | . . . . . 6 ⊢ (oppr‘𝑅) = (oppr‘𝑅) | |
| 2 | 1 | opprring 20347 | . . . . 5 ⊢ (𝑅 ∈ Ring → (oppr‘𝑅) ∈ Ring) | 
| 3 | 2 | adantr 480 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (oppr‘𝑅) ∈ Ring) | 
| 4 | simpr1 1195 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑁 ∈ ℤ) | |
| 5 | simpr3 1197 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑌 ∈ 𝐵) | |
| 6 | simpr2 1196 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝑋 ∈ 𝐵) | |
| 7 | mulgass3.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
| 8 | 1, 7 | opprbas 20341 | . . . . 5 ⊢ 𝐵 = (Base‘(oppr‘𝑅)) | 
| 9 | eqid 2737 | . . . . 5 ⊢ (.g‘(oppr‘𝑅)) = (.g‘(oppr‘𝑅)) | |
| 10 | eqid 2737 | . . . . 5 ⊢ (.r‘(oppr‘𝑅)) = (.r‘(oppr‘𝑅)) | |
| 11 | 8, 9, 10 | mulgass2 20306 | . . . 4 ⊢ (((oppr‘𝑅) ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑁(.g‘(oppr‘𝑅))𝑌)(.r‘(oppr‘𝑅))𝑋) = (𝑁(.g‘(oppr‘𝑅))(𝑌(.r‘(oppr‘𝑅))𝑋))) | 
| 12 | 3, 4, 5, 6, 11 | syl13anc 1374 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑁(.g‘(oppr‘𝑅))𝑌)(.r‘(oppr‘𝑅))𝑋) = (𝑁(.g‘(oppr‘𝑅))(𝑌(.r‘(oppr‘𝑅))𝑋))) | 
| 13 | mulgass3.t | . . . 4 ⊢ × = (.r‘𝑅) | |
| 14 | 7, 13, 1, 10 | opprmul 20337 | . . 3 ⊢ ((𝑁(.g‘(oppr‘𝑅))𝑌)(.r‘(oppr‘𝑅))𝑋) = (𝑋 × (𝑁(.g‘(oppr‘𝑅))𝑌)) | 
| 15 | 7, 13, 1, 10 | opprmul 20337 | . . . 4 ⊢ (𝑌(.r‘(oppr‘𝑅))𝑋) = (𝑋 × 𝑌) | 
| 16 | 15 | oveq2i 7442 | . . 3 ⊢ (𝑁(.g‘(oppr‘𝑅))(𝑌(.r‘(oppr‘𝑅))𝑋)) = (𝑁(.g‘(oppr‘𝑅))(𝑋 × 𝑌)) | 
| 17 | 12, 14, 16 | 3eqtr3g 2800 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 × (𝑁(.g‘(oppr‘𝑅))𝑌)) = (𝑁(.g‘(oppr‘𝑅))(𝑋 × 𝑌))) | 
| 18 | mulgass3.m | . . . . 5 ⊢ · = (.g‘𝑅) | |
| 19 | 7 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐵 = (Base‘𝑅)) | 
| 20 | 8 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐵 = (Base‘(oppr‘𝑅))) | 
| 21 | ssv 4008 | . . . . . 6 ⊢ 𝐵 ⊆ V | |
| 22 | 21 | a1i 11 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → 𝐵 ⊆ V) | 
| 23 | ovexd 7466 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘𝑅)𝑦) ∈ V) | |
| 24 | eqid 2737 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
| 25 | 1, 24 | oppradd 20343 | . . . . . . 7 ⊢ (+g‘𝑅) = (+g‘(oppr‘𝑅)) | 
| 26 | 25 | oveqi 7444 | . . . . . 6 ⊢ (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑅))𝑦) | 
| 27 | 26 | a1i 11 | . . . . 5 ⊢ (((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘(oppr‘𝑅))𝑦)) | 
| 28 | 18, 9, 19, 20, 22, 23, 27 | mulgpropd 19134 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → · = (.g‘(oppr‘𝑅))) | 
| 29 | 28 | oveqd 7448 | . . 3 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁 · 𝑌) = (𝑁(.g‘(oppr‘𝑅))𝑌)) | 
| 30 | 29 | oveq2d 7447 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑋 × (𝑁(.g‘(oppr‘𝑅))𝑌))) | 
| 31 | 28 | oveqd 7448 | . 2 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑁 · (𝑋 × 𝑌)) = (𝑁(.g‘(oppr‘𝑅))(𝑋 × 𝑌))) | 
| 32 | 17, 30, 31 | 3eqtr4d 2787 | 1 ⊢ ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 ℤcz 12613 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 .gcmg 19085 Ringcrg 20230 opprcoppr 20333 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-tpos 8251 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-seq 14043 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-plusg 17310 df-mulr 17311 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-minusg 18955 df-mulg 19086 df-cmn 19800 df-abl 19801 df-mgp 20138 df-rng 20150 df-ur 20179 df-ring 20232 df-oppr 20334 | 
| This theorem is referenced by: zlmassa 21923 psdvsca 22168 psdmul 22170 elrgspnlem2 33247 | 
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