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Theorem sraassaOLD 21890
Description: Obsolete version of sraassa 21889 as of 21-Mar-2025. (Contributed by Mario Carneiro, 6-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
sraassa.a 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
Assertion
Ref Expression
sraassaOLD ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)

Proof of Theorem sraassaOLD
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sraassa.a . . . 4 𝐴 = ((subringAlg ‘𝑊)‘𝑆)
21a1i 11 . . 3 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 = ((subringAlg ‘𝑊)‘𝑆))
3 eqid 2737 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
43subrgss 20572 . . . 4 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 ⊆ (Base‘𝑊))
54adantl 481 . . 3 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 ⊆ (Base‘𝑊))
62, 5srabase 21177 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝐴))
72, 5srasca 21183 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊s 𝑆) = (Scalar‘𝐴))
8 eqid 2737 . . . 4 (𝑊s 𝑆) = (𝑊s 𝑆)
98subrgbas 20581 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝑆 = (Base‘(𝑊s 𝑆)))
109adantl 481 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑆 = (Base‘(𝑊s 𝑆)))
112, 5sravsca 21185 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r𝑊) = ( ·𝑠𝐴))
122, 5sramulr 21181 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (.r𝑊) = (.r𝐴))
131sralmod 21194 . . 3 (𝑆 ∈ (SubRing‘𝑊) → 𝐴 ∈ LMod)
1413adantl 481 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ LMod)
15 crngring 20242 . . . 4 (𝑊 ∈ CRing → 𝑊 ∈ Ring)
1615adantr 480 . . 3 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝑊 ∈ Ring)
17 eqidd 2738 . . . 4 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (Base‘𝑊) = (Base‘𝑊))
182, 5sraaddg 21179 . . . . 5 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (+g𝑊) = (+g𝐴))
1918oveqdr 7459 . . . 4 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(+g𝑊)𝑦) = (𝑥(+g𝐴)𝑦))
2012oveqdr 7459 . . . 4 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊))) → (𝑥(.r𝑊)𝑦) = (𝑥(.r𝐴)𝑦))
2117, 6, 19, 20ringpropd 20285 . . 3 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → (𝑊 ∈ Ring ↔ 𝐴 ∈ Ring))
2216, 21mpbid 232 . 2 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ Ring)
2316adantr 480 . . 3 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑊 ∈ Ring)
245adantr 480 . . . 4 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑆 ⊆ (Base‘𝑊))
25 simpr1 1195 . . . 4 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥𝑆)
2624, 25sseldd 3984 . . 3 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑥 ∈ (Base‘𝑊))
27 simpr2 1196 . . 3 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑦 ∈ (Base‘𝑊))
28 simpr3 1197 . . 3 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → 𝑧 ∈ (Base‘𝑊))
29 eqid 2737 . . . 4 (.r𝑊) = (.r𝑊)
303, 29ringass 20250 . . 3 ((𝑊 ∈ Ring ∧ (𝑥 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r𝑊)𝑦)(.r𝑊)𝑧) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
3123, 26, 27, 28, 30syl13anc 1374 . 2 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → ((𝑥(.r𝑊)𝑦)(.r𝑊)𝑧) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
32 eqid 2737 . . . . 5 (mulGrp‘𝑊) = (mulGrp‘𝑊)
3332crngmgp 20238 . . . 4 (𝑊 ∈ CRing → (mulGrp‘𝑊) ∈ CMnd)
3433ad2antrr 726 . . 3 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (mulGrp‘𝑊) ∈ CMnd)
3532, 3mgpbas 20142 . . . 4 (Base‘𝑊) = (Base‘(mulGrp‘𝑊))
3632, 29mgpplusg 20141 . . . 4 (.r𝑊) = (+g‘(mulGrp‘𝑊))
3735, 36cmn12 19820 . . 3 (((mulGrp‘𝑊) ∈ CMnd ∧ (𝑦 ∈ (Base‘𝑊) ∧ 𝑥 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑦(.r𝑊)(𝑥(.r𝑊)𝑧)) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
3834, 27, 26, 28, 37syl13anc 1374 . 2 (((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) ∧ (𝑥𝑆𝑦 ∈ (Base‘𝑊) ∧ 𝑧 ∈ (Base‘𝑊))) → (𝑦(.r𝑊)(𝑥(.r𝑊)𝑧)) = (𝑥(.r𝑊)(𝑦(.r𝑊)𝑧)))
396, 7, 10, 11, 12, 14, 22, 31, 38isassad 21885 1 ((𝑊 ∈ CRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ AssAlg)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wss 3951  cfv 6561  (class class class)co 7431  Basecbs 17247  s cress 17274  +gcplusg 17297  .rcmulr 17298  CMndccmn 19798  mulGrpcmgp 20137  Ringcrg 20230  CRingccrg 20231  SubRingcsubrg 20569  LModclmod 20858  subringAlg csra 21170  AssAlgcasa 21870
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-rep 5279  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-2nd 8015  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-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-subg 19141  df-cmn 19800  df-mgp 20138  df-ur 20179  df-ring 20232  df-cring 20233  df-subrg 20570  df-lmod 20860  df-sra 21172  df-assa 21873
This theorem is referenced by: (None)
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