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Theorem mulgass3 19379
 Description: An associative property between group multiple and ring multiplication. (Contributed by Mario Carneiro, 14-Jun-2015.)
Hypotheses
Ref Expression
mulgass3.b 𝐵 = (Base‘𝑅)
mulgass3.m · = (.g𝑅)
mulgass3.t × = (.r𝑅)
Assertion
Ref Expression
mulgass3 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌)))

Proof of Theorem mulgass3
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2819 . . . . . 6 (oppr𝑅) = (oppr𝑅)
21opprring 19373 . . . . 5 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
32adantr 483 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (oppr𝑅) ∈ Ring)
4 simpr1 1189 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑁 ∈ ℤ)
5 simpr3 1191 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
6 simpr2 1190 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
7 mulgass3.b . . . . . 6 𝐵 = (Base‘𝑅)
81, 7opprbas 19371 . . . . 5 𝐵 = (Base‘(oppr𝑅))
9 eqid 2819 . . . . 5 (.g‘(oppr𝑅)) = (.g‘(oppr𝑅))
10 eqid 2819 . . . . 5 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
118, 9, 10mulgass2 19343 . . . 4 (((oppr𝑅) ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑌𝐵𝑋𝐵)) → ((𝑁(.g‘(oppr𝑅))𝑌)(.r‘(oppr𝑅))𝑋) = (𝑁(.g‘(oppr𝑅))(𝑌(.r‘(oppr𝑅))𝑋)))
123, 4, 5, 6, 11syl13anc 1367 . . 3 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → ((𝑁(.g‘(oppr𝑅))𝑌)(.r‘(oppr𝑅))𝑋) = (𝑁(.g‘(oppr𝑅))(𝑌(.r‘(oppr𝑅))𝑋)))
13 mulgass3.t . . . 4 × = (.r𝑅)
147, 13, 1, 10opprmul 19368 . . 3 ((𝑁(.g‘(oppr𝑅))𝑌)(.r‘(oppr𝑅))𝑋) = (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌))
157, 13, 1, 10opprmul 19368 . . . 4 (𝑌(.r‘(oppr𝑅))𝑋) = (𝑋 × 𝑌)
1615oveq2i 7159 . . 3 (𝑁(.g‘(oppr𝑅))(𝑌(.r‘(oppr𝑅))𝑋)) = (𝑁(.g‘(oppr𝑅))(𝑋 × 𝑌))
1712, 14, 163eqtr3g 2877 . 2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌)) = (𝑁(.g‘(oppr𝑅))(𝑋 × 𝑌)))
18 mulgass3.m . . . . 5 · = (.g𝑅)
197a1i 11 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝐵 = (Base‘𝑅))
208a1i 11 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝐵 = (Base‘(oppr𝑅)))
21 ssv 3989 . . . . . 6 𝐵 ⊆ V
2221a1i 11 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝐵 ⊆ V)
23 ovexd 7183 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑅)𝑦) ∈ V)
24 eqid 2819 . . . . . . . 8 (+g𝑅) = (+g𝑅)
251, 24oppradd 19372 . . . . . . 7 (+g𝑅) = (+g‘(oppr𝑅))
2625oveqi 7161 . . . . . 6 (𝑥(+g𝑅)𝑦) = (𝑥(+g‘(oppr𝑅))𝑦)
2726a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g‘(oppr𝑅))𝑦))
2818, 9, 19, 20, 22, 23, 27mulgpropd 18261 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → · = (.g‘(oppr𝑅)))
2928oveqd 7165 . . 3 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁 · 𝑌) = (𝑁(.g‘(oppr𝑅))𝑌))
3029oveq2d 7164 . 2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌)))
3128oveqd 7165 . 2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁 · (𝑋 × 𝑌)) = (𝑁(.g‘(oppr𝑅))(𝑋 × 𝑌)))
3217, 30, 313eqtr4d 2864 1 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108  Vcvv 3493   ⊆ wss 3934  ‘cfv 6348  (class class class)co 7148  ℤcz 11973  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  .gcmg 18216  Ringcrg 19289  opprcoppr 19364 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-nel 3122  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-tpos 7884  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-nn 11631  df-2 11692  df-3 11693  df-n0 11890  df-z 11974  df-uz 12236  df-fz 12885  df-seq 13362  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-plusg 16570  df-mulr 16571  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098  df-minusg 18099  df-mulg 18217  df-mgp 19232  df-ur 19244  df-ring 19291  df-oppr 19365 This theorem is referenced by:  zlmassa  20663
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