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Theorem mulgass3 18460
 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 2610 . . . . . 6 (oppr𝑅) = (oppr𝑅)
21opprring 18454 . . . . 5 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
32adantr 480 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (oppr𝑅) ∈ Ring)
4 simpr1 1060 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑁 ∈ ℤ)
5 simpr3 1062 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
6 simpr2 1061 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
7 mulgass3.b . . . . . 6 𝐵 = (Base‘𝑅)
81, 7opprbas 18452 . . . . 5 𝐵 = (Base‘(oppr𝑅))
9 eqid 2610 . . . . 5 (.g‘(oppr𝑅)) = (.g‘(oppr𝑅))
10 eqid 2610 . . . . 5 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
118, 9, 10mulgass2 18424 . . . 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 1320 . . 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 18449 . . 3 ((𝑁(.g‘(oppr𝑅))𝑌)(.r‘(oppr𝑅))𝑋) = (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌))
157, 13, 1, 10opprmul 18449 . . . 4 (𝑌(.r‘(oppr𝑅))𝑋) = (𝑋 × 𝑌)
1615oveq2i 6560 . . 3 (𝑁(.g‘(oppr𝑅))(𝑌(.r‘(oppr𝑅))𝑋)) = (𝑁(.g‘(oppr𝑅))(𝑋 × 𝑌))
1712, 14, 163eqtr3g 2667 . 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 3588 . . . . . 6 𝐵 ⊆ V
2221a1i 11 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝐵 ⊆ V)
23 ovex 6577 . . . . . 6 (𝑥(+g𝑅)𝑦) ∈ V
2423a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑅)𝑦) ∈ V)
25 eqid 2610 . . . . . . . 8 (+g𝑅) = (+g𝑅)
261, 25oppradd 18453 . . . . . . 7 (+g𝑅) = (+g‘(oppr𝑅))
2726oveqi 6562 . . . . . 6 (𝑥(+g𝑅)𝑦) = (𝑥(+g‘(oppr𝑅))𝑦)
2827a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g‘(oppr𝑅))𝑦))
2918, 9, 19, 20, 22, 24, 28mulgpropd 17407 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → · = (.g‘(oppr𝑅)))
3029oveqd 6566 . . 3 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁 · 𝑌) = (𝑁(.g‘(oppr𝑅))𝑌))
3130oveq2d 6565 . 2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌)))
3229oveqd 6566 . 2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁 · (𝑋 × 𝑌)) = (𝑁(.g‘(oppr𝑅))(𝑋 × 𝑌)))
3317, 31, 323eqtr4d 2654 1 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  ℤcz 11254  Basecbs 15695  +gcplusg 15768  .rcmulr 15769  .gcmg 17363  Ringcrg 18370  opprcoppr 18445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-tpos 7239  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-seq 12664  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-plusg 15781  df-mulr 15782  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-mulg 17364  df-mgp 18313  df-ur 18325  df-ring 18372  df-oppr 18446 This theorem is referenced by:  zlmassa  19691
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