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Theorem sraassab 21808
Description: A subring algebra is an associative algebra if and only if the subring is included in the ring's center. (Contributed by SN, 21-Mar-2025.)
Hypotheses
Ref Expression
sraassab.a 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†)
sraassab.z 𝑍 = (Cntrβ€˜(mulGrpβ€˜π‘Š))
sraassab.w (πœ‘ β†’ π‘Š ∈ Ring)
sraassab.s (πœ‘ β†’ 𝑆 ∈ (SubRingβ€˜π‘Š))
Assertion
Ref Expression
sraassab (πœ‘ β†’ (𝐴 ∈ AssAlg ↔ 𝑆 βŠ† 𝑍))

Proof of Theorem sraassab
Dummy variables π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sraassab.s . . . . . . . 8 (πœ‘ β†’ 𝑆 ∈ (SubRingβ€˜π‘Š))
2 eqid 2728 . . . . . . . . 9 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
32subrgss 20518 . . . . . . . 8 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
41, 3syl 17 . . . . . . 7 (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
54adantr 479 . . . . . 6 ((πœ‘ ∧ 𝐴 ∈ AssAlg) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
65sselda 3982 . . . . 5 (((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
7 simpllr 774 . . . . . . . 8 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ 𝐴 ∈ AssAlg)
8 eqid 2728 . . . . . . . . . . . . . 14 (π‘Š β†Ύs 𝑆) = (π‘Š β†Ύs 𝑆)
98subrgbas 20527 . . . . . . . . . . . . 13 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝑆 = (Baseβ€˜(π‘Š β†Ύs 𝑆)))
101, 9syl 17 . . . . . . . . . . . 12 (πœ‘ β†’ 𝑆 = (Baseβ€˜(π‘Š β†Ύs 𝑆)))
11 sraassab.a . . . . . . . . . . . . . . 15 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†)
1211a1i 11 . . . . . . . . . . . . . 14 (πœ‘ β†’ 𝐴 = ((subringAlg β€˜π‘Š)β€˜π‘†))
1312, 4srasca 21076 . . . . . . . . . . . . 13 (πœ‘ β†’ (π‘Š β†Ύs 𝑆) = (Scalarβ€˜π΄))
1413fveq2d 6906 . . . . . . . . . . . 12 (πœ‘ β†’ (Baseβ€˜(π‘Š β†Ύs 𝑆)) = (Baseβ€˜(Scalarβ€˜π΄)))
1510, 14eqtrd 2768 . . . . . . . . . . 11 (πœ‘ β†’ 𝑆 = (Baseβ€˜(Scalarβ€˜π΄)))
1615eqimssd 4038 . . . . . . . . . 10 (πœ‘ β†’ 𝑆 βŠ† (Baseβ€˜(Scalarβ€˜π΄)))
1716sselda 3982 . . . . . . . . 9 ((πœ‘ ∧ 𝑦 ∈ 𝑆) β†’ 𝑦 ∈ (Baseβ€˜(Scalarβ€˜π΄)))
1817ad4ant13 749 . . . . . . . 8 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ 𝑦 ∈ (Baseβ€˜(Scalarβ€˜π΄)))
1912, 4srabase 21070 . . . . . . . . . . 11 (πœ‘ β†’ (Baseβ€˜π‘Š) = (Baseβ€˜π΄))
2019eqimssd 4038 . . . . . . . . . 10 (πœ‘ β†’ (Baseβ€˜π‘Š) βŠ† (Baseβ€˜π΄))
2120ad2antrr 724 . . . . . . . . 9 (((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) β†’ (Baseβ€˜π‘Š) βŠ† (Baseβ€˜π΄))
2221sselda 3982 . . . . . . . 8 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ π‘₯ ∈ (Baseβ€˜π΄))
23 sraassab.w . . . . . . . . . . 11 (πœ‘ β†’ π‘Š ∈ Ring)
24 eqid 2728 . . . . . . . . . . . 12 (1rβ€˜π‘Š) = (1rβ€˜π‘Š)
252, 24ringidcl 20209 . . . . . . . . . . 11 (π‘Š ∈ Ring β†’ (1rβ€˜π‘Š) ∈ (Baseβ€˜π‘Š))
2623, 25syl 17 . . . . . . . . . 10 (πœ‘ β†’ (1rβ€˜π‘Š) ∈ (Baseβ€˜π‘Š))
2726, 19eleqtrd 2831 . . . . . . . . 9 (πœ‘ β†’ (1rβ€˜π‘Š) ∈ (Baseβ€˜π΄))
2827ad3antrrr 728 . . . . . . . 8 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (1rβ€˜π‘Š) ∈ (Baseβ€˜π΄))
29 eqid 2728 . . . . . . . . 9 (Baseβ€˜π΄) = (Baseβ€˜π΄)
30 eqid 2728 . . . . . . . . 9 (Scalarβ€˜π΄) = (Scalarβ€˜π΄)
31 eqid 2728 . . . . . . . . 9 (Baseβ€˜(Scalarβ€˜π΄)) = (Baseβ€˜(Scalarβ€˜π΄))
32 eqid 2728 . . . . . . . . 9 ( ·𝑠 β€˜π΄) = ( ·𝑠 β€˜π΄)
33 eqid 2728 . . . . . . . . 9 (.rβ€˜π΄) = (.rβ€˜π΄)
3429, 30, 31, 32, 33assaassr 21800 . . . . . . . 8 ((𝐴 ∈ AssAlg ∧ (𝑦 ∈ (Baseβ€˜(Scalarβ€˜π΄)) ∧ π‘₯ ∈ (Baseβ€˜π΄) ∧ (1rβ€˜π‘Š) ∈ (Baseβ€˜π΄))) β†’ (π‘₯(.rβ€˜π΄)(𝑦( ·𝑠 β€˜π΄)(1rβ€˜π‘Š))) = (𝑦( ·𝑠 β€˜π΄)(π‘₯(.rβ€˜π΄)(1rβ€˜π‘Š))))
357, 18, 22, 28, 34syl13anc 1369 . . . . . . 7 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π΄)(𝑦( ·𝑠 β€˜π΄)(1rβ€˜π‘Š))) = (𝑦( ·𝑠 β€˜π΄)(π‘₯(.rβ€˜π΄)(1rβ€˜π‘Š))))
3612, 4sramulr 21074 . . . . . . . . . 10 (πœ‘ β†’ (.rβ€˜π‘Š) = (.rβ€˜π΄))
3736ad3antrrr 728 . . . . . . . . 9 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (.rβ€˜π‘Š) = (.rβ€˜π΄))
3837oveqd 7443 . . . . . . . 8 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π‘Š)(𝑦( ·𝑠 β€˜π΄)(1rβ€˜π‘Š))) = (π‘₯(.rβ€˜π΄)(𝑦( ·𝑠 β€˜π΄)(1rβ€˜π‘Š))))
3912, 4sravsca 21078 . . . . . . . . . . . 12 (πœ‘ β†’ (.rβ€˜π‘Š) = ( ·𝑠 β€˜π΄))
4039ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (.rβ€˜π‘Š) = ( ·𝑠 β€˜π΄))
4140oveqd 7443 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (𝑦(.rβ€˜π‘Š)(1rβ€˜π‘Š)) = (𝑦( ·𝑠 β€˜π΄)(1rβ€˜π‘Š)))
42 eqid 2728 . . . . . . . . . . 11 (.rβ€˜π‘Š) = (.rβ€˜π‘Š)
4323ad3antrrr 728 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ π‘Š ∈ Ring)
446adantr 479 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
452, 42, 24, 43, 44ringridmd 20216 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (𝑦(.rβ€˜π‘Š)(1rβ€˜π‘Š)) = 𝑦)
4641, 45eqtr3d 2770 . . . . . . . . 9 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (𝑦( ·𝑠 β€˜π΄)(1rβ€˜π‘Š)) = 𝑦)
4746oveq2d 7442 . . . . . . . 8 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π‘Š)(𝑦( ·𝑠 β€˜π΄)(1rβ€˜π‘Š))) = (π‘₯(.rβ€˜π‘Š)𝑦))
4838, 47eqtr3d 2770 . . . . . . 7 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π΄)(𝑦( ·𝑠 β€˜π΄)(1rβ€˜π‘Š))) = (π‘₯(.rβ€˜π‘Š)𝑦))
4940oveqd 7443 . . . . . . . 8 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (𝑦(.rβ€˜π‘Š)(π‘₯(.rβ€˜π΄)(1rβ€˜π‘Š))) = (𝑦( ·𝑠 β€˜π΄)(π‘₯(.rβ€˜π΄)(1rβ€˜π‘Š))))
5037oveqd 7443 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π‘Š)(1rβ€˜π‘Š)) = (π‘₯(.rβ€˜π΄)(1rβ€˜π‘Š)))
51 simpr 483 . . . . . . . . . . 11 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
522, 42, 24, 43, 51ringridmd 20216 . . . . . . . . . 10 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π‘Š)(1rβ€˜π‘Š)) = π‘₯)
5350, 52eqtr3d 2770 . . . . . . . . 9 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π΄)(1rβ€˜π‘Š)) = π‘₯)
5453oveq2d 7442 . . . . . . . 8 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (𝑦(.rβ€˜π‘Š)(π‘₯(.rβ€˜π΄)(1rβ€˜π‘Š))) = (𝑦(.rβ€˜π‘Š)π‘₯))
5549, 54eqtr3d 2770 . . . . . . 7 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (𝑦( ·𝑠 β€˜π΄)(π‘₯(.rβ€˜π΄)(1rβ€˜π‘Š))) = (𝑦(.rβ€˜π‘Š)π‘₯))
5635, 48, 553eqtr3rd 2777 . . . . . 6 ((((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) ∧ π‘₯ ∈ (Baseβ€˜π‘Š)) β†’ (𝑦(.rβ€˜π‘Š)π‘₯) = (π‘₯(.rβ€˜π‘Š)𝑦))
5756ralrimiva 3143 . . . . 5 (((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) β†’ βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)(𝑦(.rβ€˜π‘Š)π‘₯) = (π‘₯(.rβ€˜π‘Š)𝑦))
58 eqid 2728 . . . . . . 7 (mulGrpβ€˜π‘Š) = (mulGrpβ€˜π‘Š)
5958, 2mgpbas 20087 . . . . . 6 (Baseβ€˜π‘Š) = (Baseβ€˜(mulGrpβ€˜π‘Š))
6058, 42mgpplusg 20085 . . . . . 6 (.rβ€˜π‘Š) = (+gβ€˜(mulGrpβ€˜π‘Š))
61 sraassab.z . . . . . 6 𝑍 = (Cntrβ€˜(mulGrpβ€˜π‘Š))
6259, 60, 61elcntr 19288 . . . . 5 (𝑦 ∈ 𝑍 ↔ (𝑦 ∈ (Baseβ€˜π‘Š) ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘Š)(𝑦(.rβ€˜π‘Š)π‘₯) = (π‘₯(.rβ€˜π‘Š)𝑦)))
636, 57, 62sylanbrc 581 . . . 4 (((πœ‘ ∧ 𝐴 ∈ AssAlg) ∧ 𝑦 ∈ 𝑆) β†’ 𝑦 ∈ 𝑍)
6463ex 411 . . 3 ((πœ‘ ∧ 𝐴 ∈ AssAlg) β†’ (𝑦 ∈ 𝑆 β†’ 𝑦 ∈ 𝑍))
6564ssrdv 3988 . 2 ((πœ‘ ∧ 𝐴 ∈ AssAlg) β†’ 𝑆 βŠ† 𝑍)
6619adantr 479 . . 3 ((πœ‘ ∧ 𝑆 βŠ† 𝑍) β†’ (Baseβ€˜π‘Š) = (Baseβ€˜π΄))
6713adantr 479 . . 3 ((πœ‘ ∧ 𝑆 βŠ† 𝑍) β†’ (π‘Š β†Ύs 𝑆) = (Scalarβ€˜π΄))
6810adantr 479 . . 3 ((πœ‘ ∧ 𝑆 βŠ† 𝑍) β†’ 𝑆 = (Baseβ€˜(π‘Š β†Ύs 𝑆)))
6939adantr 479 . . 3 ((πœ‘ ∧ 𝑆 βŠ† 𝑍) β†’ (.rβ€˜π‘Š) = ( ·𝑠 β€˜π΄))
7036adantr 479 . . 3 ((πœ‘ ∧ 𝑆 βŠ† 𝑍) β†’ (.rβ€˜π‘Š) = (.rβ€˜π΄))
7111sralmod 21087 . . . . 5 (𝑆 ∈ (SubRingβ€˜π‘Š) β†’ 𝐴 ∈ LMod)
721, 71syl 17 . . . 4 (πœ‘ β†’ 𝐴 ∈ LMod)
7372adantr 479 . . 3 ((πœ‘ ∧ 𝑆 βŠ† 𝑍) β†’ 𝐴 ∈ LMod)
7411, 2sraring 21086 . . . . 5 ((π‘Š ∈ Ring ∧ 𝑆 βŠ† (Baseβ€˜π‘Š)) β†’ 𝐴 ∈ Ring)
7523, 4, 74syl2anc 582 . . . 4 (πœ‘ β†’ 𝐴 ∈ Ring)
7675adantr 479 . . 3 ((πœ‘ ∧ 𝑆 βŠ† 𝑍) β†’ 𝐴 ∈ Ring)
7723ad2antrr 724 . . . 4 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘Š ∈ Ring)
784adantr 479 . . . . . 6 ((πœ‘ ∧ 𝑆 βŠ† 𝑍) β†’ 𝑆 βŠ† (Baseβ€˜π‘Š))
7978sselda 3982 . . . . 5 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
80793ad2antr1 1185 . . . 4 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ π‘₯ ∈ (Baseβ€˜π‘Š))
81 simpr2 1192 . . . 4 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
82 simpr3 1193 . . . 4 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ 𝑧 ∈ (Baseβ€˜π‘Š))
832, 42, 77, 80, 81, 82ringassd 20204 . . 3 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 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)))
84 ssel2 3977 . . . . . . . 8 ((𝑆 βŠ† 𝑍 ∧ π‘₯ ∈ 𝑆) β†’ π‘₯ ∈ 𝑍)
8584ad2ant2lr 746 . . . . . . 7 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š))) β†’ π‘₯ ∈ 𝑍)
86 simprr 771 . . . . . . 7 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š))) β†’ 𝑦 ∈ (Baseβ€˜π‘Š))
8759, 60, 61cntri 19290 . . . . . . 7 ((π‘₯ ∈ 𝑍 ∧ 𝑦 ∈ (Baseβ€˜π‘Š)) β†’ (π‘₯(.rβ€˜π‘Š)𝑦) = (𝑦(.rβ€˜π‘Š)π‘₯))
8885, 86, 87syl2anc 582 . . . . . 6 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š))) β†’ (π‘₯(.rβ€˜π‘Š)𝑦) = (𝑦(.rβ€˜π‘Š)π‘₯))
89883adantr3 1168 . . . . 5 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ (π‘₯(.rβ€˜π‘Š)𝑦) = (𝑦(.rβ€˜π‘Š)π‘₯))
9089oveq1d 7441 . . . 4 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ ((π‘₯(.rβ€˜π‘Š)𝑦)(.rβ€˜π‘Š)𝑧) = ((𝑦(.rβ€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑧))
912, 42, 77, 81, 80, 82ringassd 20204 . . . 4 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ ((𝑦(.rβ€˜π‘Š)π‘₯)(.rβ€˜π‘Š)𝑧) = (𝑦(.rβ€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑧)))
9290, 83, 913eqtr3rd 2777 . . 3 (((πœ‘ ∧ 𝑆 βŠ† 𝑍) ∧ (π‘₯ ∈ 𝑆 ∧ 𝑦 ∈ (Baseβ€˜π‘Š) ∧ 𝑧 ∈ (Baseβ€˜π‘Š))) β†’ (𝑦(.rβ€˜π‘Š)(π‘₯(.rβ€˜π‘Š)𝑧)) = (π‘₯(.rβ€˜π‘Š)(𝑦(.rβ€˜π‘Š)𝑧)))
9366, 67, 68, 69, 70, 73, 76, 83, 92isassad 21805 . 2 ((πœ‘ ∧ 𝑆 βŠ† 𝑍) β†’ 𝐴 ∈ AssAlg)
9465, 93impbida 799 1 (πœ‘ β†’ (𝐴 ∈ AssAlg ↔ 𝑆 βŠ† 𝑍))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058   βŠ† wss 3949  β€˜cfv 6553  (class class class)co 7426  Basecbs 17187   β†Ύs cress 17216  .rcmulr 17241  Scalarcsca 17243   ·𝑠 cvsca 17244  Cntrccntr 19274  mulGrpcmgp 20081  1rcur 20128  Ringcrg 20180  SubRingcsubrg 20513  LModclmod 20750  subringAlg csra 21063  AssAlgcasa 21791
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-3 12314  df-4 12315  df-5 12316  df-6 12317  df-7 12318  df-8 12319  df-sets 17140  df-slot 17158  df-ndx 17170  df-base 17188  df-ress 17217  df-plusg 17253  df-mulr 17254  df-sca 17256  df-vsca 17257  df-ip 17258  df-0g 17430  df-mgm 18607  df-sgrp 18686  df-mnd 18702  df-grp 18900  df-subg 19085  df-cntz 19275  df-cntr 19276  df-mgp 20082  df-ur 20129  df-ring 20182  df-subrg 20515  df-lmod 20752  df-sra 21065  df-assa 21794
This theorem is referenced by:  sraassa  21809
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