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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  resssra Structured version   Visualization version   GIF version
Theorem resssra 32962
Description: The subring algebra of a restricted structure is the restriction of the subring algebra. (Contributed by Thierry Arnoux, 2-Apr-2025.)
Hypotheses
Ref Expression
resssra.a 𝐴 = (Baseβ€˜π‘…)
resssra.s 𝑆 = (𝑅 β†Ύs 𝐡)
resssra.b (πœ‘ β†’ 𝐡 βŠ† 𝐴)
resssra.c (πœ‘ β†’ 𝐢 βŠ† 𝐡)
resssra.r (πœ‘ β†’ 𝑅 ∈ 𝑉)
Assertion
Ref Expression
resssra (πœ‘ β†’ ((subringAlg β€˜π‘†)β€˜πΆ) = (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡))
Proof of Theorem resssra
StepHypRef Expression
1 resssra.a . . . . . . 7 𝐴 = (Baseβ€˜π‘…)
2 eqidd 2731 . . . . . . . 8 (πœ‘ β†’ ((subringAlg β€˜π‘…)β€˜πΆ) = ((subringAlg β€˜π‘…)β€˜πΆ))
3 resssra.c . . . . . . . . . 10 (πœ‘ β†’ 𝐢 βŠ† 𝐡)
4 resssra.b . . . . . . . . . 10 (πœ‘ β†’ 𝐡 βŠ† 𝐴)
53, 4sstrd 3991 . . . . . . . . 9 (πœ‘ β†’ 𝐢 βŠ† 𝐴)
65, 1sseqtrdi 4031 . . . . . . . 8 (πœ‘ β†’ 𝐢 βŠ† (Baseβ€˜π‘…))
72, 6srabase 20937 . . . . . . 7 (πœ‘ β†’ (Baseβ€˜π‘…) = (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)))
81, 7eqtrid 2782 . . . . . 6 (πœ‘ β†’ 𝐴 = (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)))
98oveq2d 7427 . . . . 5 (πœ‘ β†’ (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐴) = (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ))))
109adantr 479 . . . 4 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐴) = (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ))))
11 simpr 483 . . . . . 6 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ 𝐴 βŠ† 𝐡)
124adantr 479 . . . . . 6 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ 𝐡 βŠ† 𝐴)
1311, 12eqssd 3998 . . . . 5 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ 𝐴 = 𝐡)
1413oveq2d 7427 . . . 4 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐴) = (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡))
15 fvex 6903 . . . . 5 ((subringAlg β€˜π‘…)β€˜πΆ) ∈ V
16 eqid 2730 . . . . . 6 (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)) = (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ))
1716ressid 17193 . . . . 5 (((subringAlg β€˜π‘…)β€˜πΆ) ∈ V β†’ (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ))) = ((subringAlg β€˜π‘…)β€˜πΆ))
1815, 17mp1i 13 . . . 4 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ))) = ((subringAlg β€˜π‘…)β€˜πΆ))
1910, 14, 183eqtr3d 2778 . . 3 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡) = ((subringAlg β€˜π‘…)β€˜πΆ))
201oveq2i 7422 . . . . . . . 8 (𝑅 β†Ύs 𝐴) = (𝑅 β†Ύs (Baseβ€˜π‘…))
21 resssra.r . . . . . . . . . 10 (πœ‘ β†’ 𝑅 ∈ 𝑉)
2221elexd 3493 . . . . . . . . 9 (πœ‘ β†’ 𝑅 ∈ V)
23 eqid 2730 . . . . . . . . . 10 (Baseβ€˜π‘…) = (Baseβ€˜π‘…)
2423ressid 17193 . . . . . . . . 9 (𝑅 ∈ V β†’ (𝑅 β†Ύs (Baseβ€˜π‘…)) = 𝑅)
2522, 24syl 17 . . . . . . . 8 (πœ‘ β†’ (𝑅 β†Ύs (Baseβ€˜π‘…)) = 𝑅)
2620, 25eqtrid 2782 . . . . . . 7 (πœ‘ β†’ (𝑅 β†Ύs 𝐴) = 𝑅)
2726adantr 479 . . . . . 6 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ (𝑅 β†Ύs 𝐴) = 𝑅)
2813oveq2d 7427 . . . . . . 7 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ (𝑅 β†Ύs 𝐴) = (𝑅 β†Ύs 𝐡))
29 resssra.s . . . . . . 7 𝑆 = (𝑅 β†Ύs 𝐡)
3028, 29eqtr4di 2788 . . . . . 6 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ (𝑅 β†Ύs 𝐴) = 𝑆)
3127, 30eqtr3d 2772 . . . . 5 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ 𝑅 = 𝑆)
3231fveq2d 6894 . . . 4 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ (subringAlg β€˜π‘…) = (subringAlg β€˜π‘†))
3332fveq1d 6892 . . 3 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ ((subringAlg β€˜π‘…)β€˜πΆ) = ((subringAlg β€˜π‘†)β€˜πΆ))
3419, 33eqtr2d 2771 . 2 ((πœ‘ ∧ 𝐴 βŠ† 𝐡) β†’ ((subringAlg β€˜π‘†)β€˜πΆ) = (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡))
35 simpr 483 . . . . . . . . . . 11 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ Β¬ 𝐴 βŠ† 𝐡)
3622adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ 𝑅 ∈ V)
371fvexi 6904 . . . . . . . . . . . . . 14 𝐴 ∈ V
3837a1i 11 . . . . . . . . . . . . 13 (πœ‘ β†’ 𝐴 ∈ V)
3938, 4ssexd 5323 . . . . . . . . . . . 12 (πœ‘ β†’ 𝐡 ∈ V)
4039adantr 479 . . . . . . . . . . 11 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ 𝐡 ∈ V)
4129, 1ressval2 17182 . . . . . . . . . . 11 ((Β¬ 𝐴 βŠ† 𝐡 ∧ 𝑅 ∈ V ∧ 𝐡 ∈ V) β†’ 𝑆 = (𝑅 sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ 𝐴)⟩))
4235, 36, 40, 41syl3anc 1369 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ 𝑆 = (𝑅 sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ 𝐴)⟩))
43 df-ss 3964 . . . . . . . . . . . . . 14 (𝐡 βŠ† 𝐴 ↔ (𝐡 ∩ 𝐴) = 𝐡)
444, 43sylib 217 . . . . . . . . . . . . 13 (πœ‘ β†’ (𝐡 ∩ 𝐴) = 𝐡)
4544opeq2d 4879 . . . . . . . . . . . 12 (πœ‘ β†’ ⟨(Baseβ€˜ndx), (𝐡 ∩ 𝐴)⟩ = ⟨(Baseβ€˜ndx), 𝐡⟩)
4645oveq2d 7427 . . . . . . . . . . 11 (πœ‘ β†’ (𝑅 sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ 𝐴)⟩) = (𝑅 sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
4746adantr 479 . . . . . . . . . 10 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (𝑅 sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ 𝐴)⟩) = (𝑅 sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
4842, 47eqtrd 2770 . . . . . . . . 9 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ 𝑆 = (𝑅 sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
4929oveq1i 7421 . . . . . . . . . . . 12 (𝑆 β†Ύs 𝐢) = ((𝑅 β†Ύs 𝐡) β†Ύs 𝐢)
50 ressabs 17198 . . . . . . . . . . . . 13 ((𝐡 ∈ V ∧ 𝐢 βŠ† 𝐡) β†’ ((𝑅 β†Ύs 𝐡) β†Ύs 𝐢) = (𝑅 β†Ύs 𝐢))
5139, 3, 50syl2anc 582 . . . . . . . . . . . 12 (πœ‘ β†’ ((𝑅 β†Ύs 𝐡) β†Ύs 𝐢) = (𝑅 β†Ύs 𝐢))
5249, 51eqtrid 2782 . . . . . . . . . . 11 (πœ‘ β†’ (𝑆 β†Ύs 𝐢) = (𝑅 β†Ύs 𝐢))
5352opeq2d 4879 . . . . . . . . . 10 (πœ‘ β†’ ⟨(Scalarβ€˜ndx), (𝑆 β†Ύs 𝐢)⟩ = ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩)
5453adantr 479 . . . . . . . . 9 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ ⟨(Scalarβ€˜ndx), (𝑆 β†Ύs 𝐢)⟩ = ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩)
5548, 54oveq12d 7429 . . . . . . . 8 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (𝑆 sSet ⟨(Scalarβ€˜ndx), (𝑆 β†Ύs 𝐢)⟩) = ((𝑅 sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩))
56 scandxnbasendx 17265 . . . . . . . . . . 11 (Scalarβ€˜ndx) β‰  (Baseβ€˜ndx)
5756a1i 11 . . . . . . . . . 10 (πœ‘ β†’ (Scalarβ€˜ndx) β‰  (Baseβ€˜ndx))
58 ovexd 7446 . . . . . . . . . 10 (πœ‘ β†’ (𝑅 β†Ύs 𝐢) ∈ V)
59 fvex 6903 . . . . . . . . . . 11 (Scalarβ€˜ndx) ∈ V
60 fvex 6903 . . . . . . . . . . 11 (Baseβ€˜ndx) ∈ V
6159, 60setscom 17117 . . . . . . . . . 10 (((𝑅 ∈ V ∧ (Scalarβ€˜ndx) β‰  (Baseβ€˜ndx)) ∧ ((𝑅 β†Ύs 𝐢) ∈ V ∧ 𝐡 ∈ V)) β†’ ((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = ((𝑅 sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩))
6222, 57, 58, 39, 61syl22anc 835 . . . . . . . . 9 (πœ‘ β†’ ((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = ((𝑅 sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩))
6362adantr 479 . . . . . . . 8 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ ((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = ((𝑅 sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩))
6455, 63eqtr4d 2773 . . . . . . 7 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (𝑆 sSet ⟨(Scalarβ€˜ndx), (𝑆 β†Ύs 𝐢)⟩) = ((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
65 eqid 2730 . . . . . . . . . . . 12 (.rβ€˜π‘…) = (.rβ€˜π‘…)
6629, 65ressmulr 17256 . . . . . . . . . . 11 (𝐡 ∈ V β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
6739, 66syl 17 . . . . . . . . . 10 (πœ‘ β†’ (.rβ€˜π‘…) = (.rβ€˜π‘†))
6867eqcomd 2736 . . . . . . . . 9 (πœ‘ β†’ (.rβ€˜π‘†) = (.rβ€˜π‘…))
6968opeq2d 4879 . . . . . . . 8 (πœ‘ β†’ ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘†)⟩ = ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩)
7069adantr 479 . . . . . . 7 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘†)⟩ = ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩)
7164, 70oveq12d 7429 . . . . . 6 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ ((𝑆 sSet ⟨(Scalarβ€˜ndx), (𝑆 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘†)⟩) = (((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩))
72 ovexd 7446 . . . . . . . 8 (πœ‘ β†’ (𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) ∈ V)
73 vscandxnbasendx 17270 . . . . . . . . 9 ( ·𝑠 β€˜ndx) β‰  (Baseβ€˜ndx)
7473a1i 11 . . . . . . . 8 (πœ‘ β†’ ( ·𝑠 β€˜ndx) β‰  (Baseβ€˜ndx))
75 fvexd 6905 . . . . . . . 8 (πœ‘ β†’ (.rβ€˜π‘…) ∈ V)
76 fvex 6903 . . . . . . . . 9 ( ·𝑠 β€˜ndx) ∈ V
7776, 60setscom 17117 . . . . . . . 8 ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) ∈ V ∧ ( ·𝑠 β€˜ndx) β‰  (Baseβ€˜ndx)) ∧ ((.rβ€˜π‘…) ∈ V ∧ 𝐡 ∈ V)) β†’ (((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = (((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩))
7872, 74, 75, 39, 77syl22anc 835 . . . . . . 7 (πœ‘ β†’ (((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = (((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩))
7978adantr 479 . . . . . 6 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = (((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩))
8071, 79eqtr4d 2773 . . . . 5 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ ((𝑆 sSet ⟨(Scalarβ€˜ndx), (𝑆 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘†)⟩) = (((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
8168opeq2d 4879 . . . . . 6 (πœ‘ β†’ ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘†)⟩ = ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩)
8281adantr 479 . . . . 5 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘†)⟩ = ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩)
8380, 82oveq12d 7429 . . . 4 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (((𝑆 sSet ⟨(Scalarβ€˜ndx), (𝑆 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘†)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘†)⟩) = ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩))
84 ovexd 7446 . . . . . 6 (πœ‘ β†’ ((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) ∈ V)
85 ipndxnbasendx 17281 . . . . . . 7 (Β·π‘–β€˜ndx) β‰  (Baseβ€˜ndx)
8685a1i 11 . . . . . 6 (πœ‘ β†’ (Β·π‘–β€˜ndx) β‰  (Baseβ€˜ndx))
87 fvex 6903 . . . . . . 7 (Β·π‘–β€˜ndx) ∈ V
8887, 60setscom 17117 . . . . . 6 (((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) ∈ V ∧ (Β·π‘–β€˜ndx) β‰  (Baseβ€˜ndx)) ∧ ((.rβ€˜π‘…) ∈ V ∧ 𝐡 ∈ V)) β†’ ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩))
8984, 86, 75, 39, 88syl22anc 835 . . . . 5 (πœ‘ β†’ ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩))
9089adantr 479 . . . 4 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩))
9183, 90eqtr4d 2773 . . 3 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (((𝑆 sSet ⟨(Scalarβ€˜ndx), (𝑆 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘†)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘†)⟩) = ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
9229ovexi 7445 . . . 4 𝑆 ∈ V
9329, 1ressbas2 17186 . . . . . . 7 (𝐡 βŠ† 𝐴 β†’ 𝐡 = (Baseβ€˜π‘†))
944, 93syl 17 . . . . . 6 (πœ‘ β†’ 𝐡 = (Baseβ€˜π‘†))
953, 94sseqtrd 4021 . . . . 5 (πœ‘ β†’ 𝐢 βŠ† (Baseβ€˜π‘†))
9695adantr 479 . . . 4 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ 𝐢 βŠ† (Baseβ€˜π‘†))
97 sraval 20934 . . . 4 ((𝑆 ∈ V ∧ 𝐢 βŠ† (Baseβ€˜π‘†)) β†’ ((subringAlg β€˜π‘†)β€˜πΆ) = (((𝑆 sSet ⟨(Scalarβ€˜ndx), (𝑆 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘†)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘†)⟩))
9892, 96, 97sylancr 585 . . 3 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ ((subringAlg β€˜π‘†)β€˜πΆ) = (((𝑆 sSet ⟨(Scalarβ€˜ndx), (𝑆 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘†)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘†)⟩))
998adantr 479 . . . . . . 7 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ 𝐴 = (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)))
10099sseq1d 4012 . . . . . 6 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (𝐴 βŠ† 𝐡 ↔ (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)) βŠ† 𝐡))
10135, 100mtbid 323 . . . . 5 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ Β¬ (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)) βŠ† 𝐡)
102 fvexd 6905 . . . . 5 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ ((subringAlg β€˜π‘…)β€˜πΆ) ∈ V)
103 eqid 2730 . . . . . 6 (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡) = (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡)
104103, 16ressval2 17182 . . . . 5 ((Β¬ (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)) βŠ† 𝐡 ∧ ((subringAlg β€˜π‘…)β€˜πΆ) ∈ V ∧ 𝐡 ∈ V) β†’ (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡) = (((subringAlg β€˜π‘…)β€˜πΆ) sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)))⟩))
105101, 102, 40, 104syl3anc 1369 . . . 4 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡) = (((subringAlg β€˜π‘…)β€˜πΆ) sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)))⟩))
1068ineq2d 4211 . . . . . . . 8 (πœ‘ β†’ (𝐡 ∩ 𝐴) = (𝐡 ∩ (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ))))
107106, 44eqtr3d 2772 . . . . . . 7 (πœ‘ β†’ (𝐡 ∩ (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ))) = 𝐡)
108107opeq2d 4879 . . . . . 6 (πœ‘ β†’ ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)))⟩ = ⟨(Baseβ€˜ndx), 𝐡⟩)
109108oveq2d 7427 . . . . 5 (πœ‘ β†’ (((subringAlg β€˜π‘…)β€˜πΆ) sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)))⟩) = (((subringAlg β€˜π‘…)β€˜πΆ) sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
110109adantr 479 . . . 4 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (((subringAlg β€˜π‘…)β€˜πΆ) sSet ⟨(Baseβ€˜ndx), (𝐡 ∩ (Baseβ€˜((subringAlg β€˜π‘…)β€˜πΆ)))⟩) = (((subringAlg β€˜π‘…)β€˜πΆ) sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
111 sraval 20934 . . . . . . 7 ((𝑅 ∈ 𝑉 ∧ 𝐢 βŠ† (Baseβ€˜π‘…)) β†’ ((subringAlg β€˜π‘…)β€˜πΆ) = (((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩))
11221, 6, 111syl2anc 582 . . . . . 6 (πœ‘ β†’ ((subringAlg β€˜π‘…)β€˜πΆ) = (((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩))
113112oveq1d 7426 . . . . 5 (πœ‘ β†’ (((subringAlg β€˜π‘…)β€˜πΆ) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
114113adantr 479 . . . 4 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (((subringAlg β€˜π‘…)β€˜πΆ) sSet ⟨(Baseβ€˜ndx), 𝐡⟩) = ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
115105, 110, 1143eqtrd 2774 . . 3 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡) = ((((𝑅 sSet ⟨(Scalarβ€˜ndx), (𝑅 β†Ύs 𝐢)⟩) sSet ⟨( ·𝑠 β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Β·π‘–β€˜ndx), (.rβ€˜π‘…)⟩) sSet ⟨(Baseβ€˜ndx), 𝐡⟩))
11691, 98, 1153eqtr4d 2780 . 2 ((πœ‘ ∧ Β¬ 𝐴 βŠ† 𝐡) β†’ ((subringAlg β€˜π‘†)β€˜πΆ) = (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡))
11734, 116pm2.61dan 809 1 (πœ‘ β†’ ((subringAlg β€˜π‘†)β€˜πΆ) = (((subringAlg β€˜π‘…)β€˜πΆ) β†Ύs 𝐡))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104   β‰  wne 2938  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  βŸ¨cop 4633  β€˜cfv 6542  (class class class)co 7411   sSet csts 17100  ndxcnx 17130  Basecbs 17148   β†Ύs cress 17177  .rcmulr 17202  Scalarcsca 17204   ·𝑠 cvsca 17205  Β·π‘–cip 17206  subringAlg csra 20926
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-mulr 17215  df-sca 17217  df-vsca 17218  df-ip 17219  df-sra 20930
This theorem is referenced by:  lsssra  32963  algextdeglem2  33063
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