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Theorem sralmod 13639
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 9 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))
3 eqid 2187 . . . 4 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
43subrgss 13442 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
5 subrgrcl 13446 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ π‘Š ∈ Ring)
62, 4, 5srabaseg 13628 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (Baseβ€˜π‘Š) = (Baseβ€˜π΄))
72, 4, 5sraaddgg 13629 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (+gβ€˜π‘Š) = (+gβ€˜π΄))
82, 4, 5srascag 13631 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (π‘Š β†Ύs 𝑆) = (Scalarβ€˜π΄))
92, 4, 5sravscag 13632 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (.rβ€˜π‘Š) = ( ·𝑠 β€˜π΄))
10 eqidd 2188 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (π‘Š β†Ύs 𝑆) = (π‘Š β†Ύs 𝑆))
11 eqidd 2188 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š))
12 id 19 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝑆 ∈ (SubRingβ€˜π‘Š))
1310, 11, 5, 12ressbasd 12541 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (𝑆 ∩ (Baseβ€˜π‘Š)) = (Baseβ€˜(π‘Š β†Ύs 𝑆)))
14 eqidd 2188 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (+gβ€˜π‘Š) = (+gβ€˜π‘Š))
1510, 14, 12, 5ressplusgd 12602 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (+gβ€˜π‘Š) = (+gβ€˜(π‘Š β†Ύs 𝑆)))
16 eqid 2187 . . . 4 (π‘Š β†Ύs 𝑆) = (π‘Š β†Ύs 𝑆)
17 eqid 2187 . . . 4 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
1816, 17ressmulrg 12618 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘Š ∈ Ring) β†’ (.rβ€˜π‘Š) = (.rβ€˜(π‘Š β†Ύs 𝑆)))
195, 18mpdan 421 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (.rβ€˜π‘Š) = (.rβ€˜(π‘Š β†Ύs 𝑆)))
20 eqid 2187 . . 3 (1rβ€˜π‘Š) = (1rβ€˜π‘Š)
2116, 20subrg1 13451 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (1rβ€˜π‘Š) = (1rβ€˜(π‘Š β†Ύs 𝑆)))
2216subrgring 13444 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (π‘Š β†Ύs 𝑆) ∈ Ring)
235ringgrpd 13257 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ π‘Š ∈ Grp)
247oveqdr 5916 . . . 4 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š))) β†’ (π‘₯(+gβ€˜π‘Š)𝑦) = (π‘₯(+gβ€˜π΄)𝑦))
2511, 6, 24grppropd 12915 . . 3 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ (π‘Š ∈ Grp ↔ 𝐴 ∈ Grp))
2623, 25mpbid 147 . 2 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 ∈ Grp)
2753ad2ant1 1019 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ π‘Š ∈ Ring)
28 elinel2 3334 . . . 4 (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
29283ad2ant2 1020 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
30 simp3 1000 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
313, 17ringcl 13265 . . 3 ((π‘Š ∈ Ring ∧ π‘₯ ∈ (Baseβ€˜π‘Š) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π‘Š)𝑦) ∈ (Baseβ€˜π‘Š))
3227, 29, 30, 31syl3anc 1248 . 2 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π‘Š)𝑦) ∈ (Baseβ€˜π‘Š))
335adantr 276 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘Š ∈ Ring)
34 simpr1 1004 . . . 4 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)))
3534elin2d 3337 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
36 simpr2 1005 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
37 simpr3 1006 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
38 eqid 2187 . . . 4 (+gβ€˜π‘Š) = (+gβ€˜π‘Š)
393, 38, 17ringdi 13270 . . 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)))
4033, 35, 36, 37, 39syl13anc 1250 . 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)))
415adantr 276 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘Š ∈ Ring)
42 simpr1 1004 . . . 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)))
4342elin2d 3337 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
44 simpr2 1005 . . . 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)))
4544elin2d 3337 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
46 simpr3 1006 . . 3 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ (π‘₯ ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑦 ∈ (𝑆 ∩ (Baseβ€˜π‘Š)) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
473, 38, 17ringdir 13271 . . 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)))
4841, 43, 45, 46, 47syl13anc 1250 . 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)))
493, 17ringass 13268 . . 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)))
5041, 43, 45, 46, 49syl13anc 1250 . 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)))
513, 17, 20ringlidm 13275 . . 3 ((π‘Š ∈ Ring ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ ((1rβ€˜π‘Š)(.rβ€˜π‘Š)π‘₯) = π‘₯)
525, 51sylan 283 . 2 ((𝑆 ∈ (SubRingβ€˜π‘Š) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ ((1rβ€˜π‘Š)(.rβ€˜π‘Š)π‘₯) = π‘₯)
536, 7, 8, 9, 13, 15, 19, 21, 22, 26, 32, 40, 48, 50, 52islmodd 13482 1 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 ∈ LMod)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ∧ w3a 979   = wceq 1363   ∈ wcel 2158   ∩ cin 3140  β€˜cfv 5228  (class class class)co 5888  Basecbs 12476   β†Ύs cress 12477  +gcplusg 12551  .rcmulr 12552  Grpcgrp 12899  1rcur 13211  Ringcrg 13248  SubRingcsubrg 13437  LModclmod 13476  subringAlg csra 13622
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-addass 7927  ax-i2m1 7930  ax-0lt1 7931  ax-0id 7933  ax-rnegex 7934  ax-pre-ltirr 7937  ax-pre-lttrn 7939  ax-pre-ltadd 7941
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-pnf 8008  df-mnf 8009  df-ltxr 8011  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-5 8995  df-6 8996  df-7 8997  df-8 8998  df-ndx 12479  df-slot 12480  df-base 12482  df-sets 12483  df-iress 12484  df-plusg 12564  df-mulr 12565  df-sca 12567  df-vsca 12568  df-ip 12569  df-0g 12725  df-mgm 12794  df-sgrp 12827  df-mnd 12840  df-grp 12902  df-subg 13062  df-mgp 13173  df-ur 13212  df-ring 13250  df-subrg 13439  df-lmod 13478  df-sra 13624
This theorem is referenced by:  rlmlmod  13653
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