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Theorem mulgass3 13066
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 2177 . . . . . 6 (oppr𝑅) = (oppr𝑅)
21opprring 13061 . . . . 5 (𝑅 ∈ Ring → (oppr𝑅) ∈ Ring)
32adantr 276 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (oppr𝑅) ∈ Ring)
4 simpr1 1003 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑁 ∈ ℤ)
5 simpr3 1005 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑌𝐵)
6 mulgass3.b . . . . . . 7 𝐵 = (Base‘𝑅)
71, 6opprbasg 13059 . . . . . 6 (𝑅 ∈ Ring → 𝐵 = (Base‘(oppr𝑅)))
87adantr 276 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝐵 = (Base‘(oppr𝑅)))
95, 8eleqtrd 2256 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑌 ∈ (Base‘(oppr𝑅)))
10 simpr2 1004 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑋𝐵)
1110, 8eleqtrd 2256 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑋 ∈ (Base‘(oppr𝑅)))
12 eqid 2177 . . . . 5 (Base‘(oppr𝑅)) = (Base‘(oppr𝑅))
13 eqid 2177 . . . . 5 (.g‘(oppr𝑅)) = (.g‘(oppr𝑅))
14 eqid 2177 . . . . 5 (.r‘(oppr𝑅)) = (.r‘(oppr𝑅))
1512, 13, 14mulgass2 13048 . . . 4 (((oppr𝑅) ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑌 ∈ (Base‘(oppr𝑅)) ∧ 𝑋 ∈ (Base‘(oppr𝑅)))) → ((𝑁(.g‘(oppr𝑅))𝑌)(.r‘(oppr𝑅))𝑋) = (𝑁(.g‘(oppr𝑅))(𝑌(.r‘(oppr𝑅))𝑋)))
163, 4, 9, 11, 15syl13anc 1240 . . 3 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → ((𝑁(.g‘(oppr𝑅))𝑌)(.r‘(oppr𝑅))𝑋) = (𝑁(.g‘(oppr𝑅))(𝑌(.r‘(oppr𝑅))𝑋)))
17 simpl 109 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝑅 ∈ Ring)
183ringgrpd 13001 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (oppr𝑅) ∈ Grp)
1912, 13, 18, 4, 9mulgcld 12880 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁(.g‘(oppr𝑅))𝑌) ∈ (Base‘(oppr𝑅)))
2019, 8eleqtrrd 2257 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁(.g‘(oppr𝑅))𝑌) ∈ 𝐵)
21 mulgass3.t . . . . 5 × = (.r𝑅)
226, 21, 1, 14opprmulg 13055 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁(.g‘(oppr𝑅))𝑌) ∈ 𝐵𝑋𝐵) → ((𝑁(.g‘(oppr𝑅))𝑌)(.r‘(oppr𝑅))𝑋) = (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌)))
2317, 20, 10, 22syl3anc 1238 . . 3 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → ((𝑁(.g‘(oppr𝑅))𝑌)(.r‘(oppr𝑅))𝑋) = (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌)))
246, 21, 1, 14opprmulg 13055 . . . . 5 ((𝑅 ∈ Ring ∧ 𝑌𝐵𝑋𝐵) → (𝑌(.r‘(oppr𝑅))𝑋) = (𝑋 × 𝑌))
2517, 5, 10, 24syl3anc 1238 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑌(.r‘(oppr𝑅))𝑋) = (𝑋 × 𝑌))
2625oveq2d 5884 . . 3 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁(.g‘(oppr𝑅))(𝑌(.r‘(oppr𝑅))𝑋)) = (𝑁(.g‘(oppr𝑅))(𝑋 × 𝑌)))
2716, 23, 263eqtr3d 2218 . 2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌)) = (𝑁(.g‘(oppr𝑅))(𝑋 × 𝑌)))
28 mulgass3.m . . . . . 6 · = (.g𝑅)
2928a1i 9 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → · = (.g𝑅))
30 eqidd 2178 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (.g‘(oppr𝑅)) = (.g‘(oppr𝑅)))
316a1i 9 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝐵 = (Base‘𝑅))
32 ssidd 3176 . . . . 5 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → 𝐵𝐵)
33 eqid 2177 . . . . . . . 8 (+g𝑅) = (+g𝑅)
346, 33ringacl 13026 . . . . . . 7 ((𝑅 ∈ Ring ∧ 𝑥𝐵𝑦𝐵) → (𝑥(+g𝑅)𝑦) ∈ 𝐵)
35343expb 1204 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) ∈ 𝐵)
3635adantlr 477 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) ∈ 𝐵)
371, 33oppraddg 13060 . . . . . . 7 (𝑅 ∈ Ring → (+g𝑅) = (+g‘(oppr𝑅)))
3837oveqdr 5896 . . . . . 6 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g‘(oppr𝑅))𝑦))
3938adantr 276 . . . . 5 (((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝑅)𝑦) = (𝑥(+g‘(oppr𝑅))𝑦))
4029, 30, 17, 3, 31, 8, 32, 36, 39mulgpropdg 12900 . . . 4 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → · = (.g‘(oppr𝑅)))
4140oveqd 5885 . . 3 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁 · 𝑌) = (𝑁(.g‘(oppr𝑅))𝑌))
4241oveq2d 5884 . 2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑋 × (𝑁(.g‘(oppr𝑅))𝑌)))
4340oveqd 5885 . 2 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑁 · (𝑋 × 𝑌)) = (𝑁(.g‘(oppr𝑅))(𝑋 × 𝑌)))
4427, 42, 433eqtr4d 2220 1 ((𝑅 ∈ Ring ∧ (𝑁 ∈ ℤ ∧ 𝑋𝐵𝑌𝐵)) → (𝑋 × (𝑁 · 𝑌)) = (𝑁 · (𝑋 × 𝑌)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  cfv 5211  (class class class)co 5868  cz 9229  Basecbs 12432  +gcplusg 12505  .rcmulr 12506  .gcmg 12859  Ringcrg 12992  opprcoppr 13051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-addcom 7889  ax-addass 7891  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-0id 7897  ax-rnegex 7898  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-ltadd 7905
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-ilim 4365  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-tpos 6239  df-recs 6299  df-frec 6385  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-inn 8896  df-2 8954  df-3 8955  df-n0 9153  df-z 9230  df-uz 9505  df-fz 9983  df-seqfrec 10419  df-ndx 12435  df-slot 12436  df-base 12438  df-sets 12439  df-plusg 12518  df-mulr 12519  df-0g 12642  df-mgm 12654  df-sgrp 12687  df-mnd 12697  df-grp 12757  df-minusg 12758  df-mulg 12860  df-mgp 12945  df-ur 12956  df-ring 12994  df-oppr 13052
This theorem is referenced by: (None)
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