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Theorem sralmod 21040
Description: The subring algebra is a left module. (Contributed by Stefan O'Rear, 27-Nov-2014.)
Hypothesis
Ref Expression
sralmod.a 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†)
Assertion
Ref Expression
sralmod (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 ∈ LMod)

Proof of Theorem sralmod
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sralmod.a . . . 4 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†)
21a1i 11 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))
3 eqid 2726 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
43subrgss 20471 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
52, 4srabase 21023 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (Baseβ€˜π‘Š) = (Baseβ€˜π΄))
62, 4sraaddg 21025 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (+gβ€˜π‘Š) = (+gβ€˜π΄))
72, 4srasca 21029 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (π‘Š β†Ύs 𝑆) = (Scalarβ€˜π΄))
82, 4sravsca 21031 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (.rβ€˜π‘Š) = ( ·𝑠 β€˜π΄))
9 eqid 2726 . . 3 (π‘Š β†Ύs 𝑆) = (π‘Š β†Ύs 𝑆)
109, 3ressbas 17185 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (𝑆 ∩ (Baseβ€˜π‘Š)) = (Baseβ€˜(π‘Š β†Ύs 𝑆)))
11 eqid 2726 . . 3 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
129, 11ressplusg 17241 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (+gβ€˜π‘Š) = (+gβ€˜(π‘Š β†Ύs 𝑆)))
13 eqid 2726 . . 3 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
149, 13ressmulr 17258 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (.rβ€˜π‘Š) = (.rβ€˜(π‘Š β†Ύs 𝑆)))
15 eqid 2726 . . 3 (1rβ€˜π‘Š) = (1rβ€˜π‘Š)
169, 15subrg1 20481 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (1rβ€˜π‘Š) = (1rβ€˜(π‘Š β†Ύs 𝑆)))
179subrgring 20473 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (π‘Š β†Ύs 𝑆) ∈ Ring)
18 subrgrcl 20475 . . . 4 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ π‘Š ∈ Ring)
19 ringgrp 20140 . . . 4 (π‘Š ∈ Ring β†’ π‘Š ∈ Grp)
2018, 19syl 17 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ π‘Š ∈ Grp)
21 eqidd 2727 . . . 4 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š))
226oveqdr 7432 . . . 4 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š))) β†’ (π‘₯(+gβ€˜π‘Š)𝑦) = (π‘₯(+gβ€˜π΄)𝑦))
2321, 5, 22grppropd 18878 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (π‘Š ∈ Grp ↔ 𝐴 ∈ Grp))
2420, 23mpbid 231 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 ∈ Grp)
25183ad2ant1 1130 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ π‘Š ∈ Ring)
26 elinel2 4191 . . . 4 (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
27263ad2ant2 1131 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
28 simp3 1135 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
293, 13ringcl 20152 . . 3 ((π‘Š ∈ Ring ∧ π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π‘Š)𝑦) ∈ (Baseβ€˜π‘Š))
3025, 27, 28, 29syl3anc 1368 . 2 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π‘Š)𝑦) ∈ (Baseβ€˜π‘Š))
3118adantr 480 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘Š ∈ Ring)
32 simpr1 1191 . . . 4 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)))
3332elin2d 4194 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
34 simpr2 1192 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
35 simpr3 1193 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
363, 11, 13ringdi 20160 . . 3 ((π‘Š ∈ Ring ∧ (π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ (π‘₯(.rβ€˜π‘Š)(𝑦(+gβ€˜π‘Š)𝑧)) = ((π‘₯(.rβ€˜π‘Š)𝑦)(+gβ€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑧)))
3731, 33, 34, 35, 36syl13anc 1369 . 2 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ (π‘₯(.rβ€˜π‘Š)(𝑦(+gβ€˜π‘Š)𝑧)) = ((π‘₯(.rβ€˜π‘Š)𝑦)(+gβ€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑧)))
3818adantr 480 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘Š ∈ Ring)
39 simpr1 1191 . . . 4 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)))
4039elin2d 4194 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
41 simpr2 1192 . . . 4 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)))
4241elin2d 4194 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
43 simpr3 1193 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
443, 11, 13ringdir 20161 . . 3 ((π‘Š ∈ Ring ∧ (π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ ((π‘₯(+gβ€˜π‘Š)𝑦)(.rβ€˜π‘Š)𝑧) = ((π‘₯(.rβ€˜π‘Š)𝑧)(+gβ€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
4538, 40, 42, 43, 44syl13anc 1369 . 2 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ ((π‘₯(+gβ€˜π‘Š)𝑦)(.rβ€˜π‘Š)𝑧) = ((π‘₯(.rβ€˜π‘Š)𝑧)(+gβ€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
463, 13ringass 20155 . . 3 ((π‘Š ∈ Ring ∧ (π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ ((π‘₯(.rβ€˜π‘Š)𝑦)(.rβ€˜π‘Š)𝑧) = (π‘₯(.rβ€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
4738, 40, 42, 43, 46syl13anc 1369 . 2 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ ((π‘₯(.rβ€˜π‘Š)𝑦)(.rβ€˜π‘Š)𝑧) = (π‘₯(.rβ€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
483, 13, 15ringlidm 20165 . . 3 ((π‘Š ∈ Ring ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ ((1rβ€˜π‘Š)(.rβ€˜π‘Š)π‘₯) = π‘₯)
4918, 48sylan 579 . 2 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ ((1rβ€˜π‘Š)(.rβ€˜π‘Š)π‘₯) = π‘₯)
505, 6, 7, 8, 10, 12, 14, 16, 17, 24, 30, 37, 45, 47, 49islmodd 20709 1 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 ∈ LMod)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   ∩ cin 3942  β€˜cfv 6536  (class class class)co 7404  Basecbs 17150   β†Ύs cress 17179  +gcplusg 17203  .rcmulr 17204  Grpcgrp 18860  1rcur 20083  Ringcrg 20135  SubRingcsubrg 20466  LModclmod 20703  subringAlg csra 21016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6293  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-2nd 7972  df-frecs 8264  df-wrecs 8295  df-recs 8369  df-rdg 8408  df-er 8702  df-en 8939  df-dom 8940  df-sdom 8941  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11447  df-neg 11448  df-nn 12214  df-2 12276  df-3 12277  df-4 12278  df-5 12279  df-6 12280  df-7 12281  df-8 12282  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-mulr 17217  df-sca 17219  df-vsca 17220  df-ip 17221  df-0g 17393  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-grp 18863  df-subg 19047  df-mgp 20037  df-ur 20084  df-ring 20137  df-subrg 20468  df-lmod 20705  df-sra 21018
This theorem is referenced by:  rlmlmod  21056  sraassab  21757  sraassaOLD  21759  sranlm  24551  sralvec  33189  lsssra  33192  evls1maplmhm  33278  algextdeglem2  33294
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