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Theorem mulgass3 18408
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 2609 . . . . . 6 (oppr𝑅) = (oppr𝑅)
21opprring 18402 . . . . 5 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
32adantr 479 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (oppr𝑅) ∈ Ring)
4 simpr1 1059 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑁 ∈ ℤ)
5 simpr3 1061 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
6 simpr2 1060 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
7 mulgass3.b . . . . . 6 𝐵 = (Base‘𝑅)
81, 7opprbas 18400 . . . . 5 𝐵 = (Base‘(oppr𝑅))
9 eqid 2609 . . . . 5 (.g‘(oppr𝑅)) = (.g‘(oppr𝑅))
10 eqid 2609 . . . . 5 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
118, 9, 10mulgass2 18372 . . . 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 1319 . . 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 18397 . . 3 ((𝑁(.g‘(oppr𝑅))𝑌)(.r‘(oppr𝑅))𝑋) = (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌))
157, 13, 1, 10opprmul 18397 . . . 4 (𝑌(.r‘(oppr𝑅))𝑋) = (𝑋 × 𝑌)
1615oveq2i 6537 . . 3 (𝑁(.g‘(oppr𝑅))(𝑌(.r‘(oppr𝑅))𝑋)) = (𝑁(.g‘(oppr𝑅))(𝑋 × 𝑌))
1712, 14, 163eqtr3g 2666 . 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 3587 . . . . . 6 𝐵 ⊆ V
2221a1i 11 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝐵 ⊆ V)
23 ovex 6554 . . . . . 6 (𝑥(+g𝑅)𝑦) ∈ V
2423a1i 11 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) ∧ (𝑥 ∈ V ∧ 𝑦 ∈ V)) → (𝑥(+g𝑅)𝑦) ∈ V)
25 eqid 2609 . . . . . . . 8 (+g𝑅) = (+g𝑅)
261, 25oppradd 18401 . . . . . . 7 (+g𝑅) = (+g‘(oppr𝑅))
2726oveqi 6539 . . . . . 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 17355 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → · = (.g‘(oppr𝑅)))
3029oveqd 6543 . . 3 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁 · 𝑌) = (𝑁(.g‘(oppr𝑅))𝑌))
3130oveq2d 6542 . 2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌)))
3229oveqd 6543 . 2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁 · (𝑋 × 𝑌)) = (𝑁(.g‘(oppr𝑅))(𝑋 × 𝑌)))
3317, 31, 323eqtr4d 2653 1 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030   = wceq 1474  wcel 1976  Vcvv 3172  wss 3539  cfv 5789  (class class class)co 6526  cz 11212  Basecbs 15643  +gcplusg 15716  .rcmulr 15717  .gcmg 17311  Ringcrg 18318  opprcoppr 18393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824  ax-inf2 8398  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4938  df-id 4942  df-po 4948  df-so 4949  df-fr 4986  df-we 4988  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-pred 5582  df-ord 5628  df-on 5629  df-lim 5630  df-suc 5631  df-iota 5753  df-fun 5791  df-fn 5792  df-f 5793  df-f1 5794  df-fo 5795  df-f1o 5796  df-fv 5797  df-riota 6488  df-ov 6529  df-oprab 6530  df-mpt2 6531  df-om 6935  df-1st 7036  df-2nd 7037  df-tpos 7216  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-er 7606  df-en 7819  df-dom 7820  df-sdom 7821  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-nn 10870  df-2 10928  df-3 10929  df-n0 11142  df-z 11213  df-uz 11522  df-fz 12155  df-seq 12621  df-ndx 15646  df-slot 15647  df-base 15648  df-sets 15649  df-plusg 15729  df-mulr 15730  df-0g 15873  df-mgm 17013  df-sgrp 17055  df-mnd 17066  df-grp 17196  df-minusg 17197  df-mulg 17312  df-mgp 18261  df-ur 18273  df-ring 18320  df-oppr 18394
This theorem is referenced by:  zlmassa  19638
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