Metamath Proof Explorer

Theorem resspsrmul

Description: A restricted power series algebra has the same multiplication operation. (Contributed by Mario Carneiro, 3-Jul-2015)

Ref Expression
Hypotheses resspsr.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
resspsr.h ⊢ 𝐻 = ( 𝑅 ↾s 𝑇 )
resspsr.u ⊢ 𝑈 = ( 𝐼 mPwSer 𝐻 )
resspsr.b ⊢ 𝐵 = ( Base ‘ 𝑈 )
resspsr.p ⊢ 𝑃 = ( 𝑆 ↾s 𝐵 )
resspsr.2 ⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) )
Assertion resspsrmul ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( .r ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( .r ‘ 𝑃 ) 𝑌 ) )

Proof

Step Hyp Ref Expression
1 resspsr.s ⊢ 𝑆 = ( 𝐼 mPwSer 𝑅 )
2 resspsr.h ⊢ 𝐻 = ( 𝑅 ↾s 𝑇 )
3 resspsr.u ⊢ 𝑈 = ( 𝐼 mPwSer 𝐻 )
4 resspsr.b ⊢ 𝐵 = ( Base ‘ 𝑈 )
5 resspsr.p ⊢ 𝑃 = ( 𝑆 ↾s 𝐵 )
6 resspsr.2 ⊢ ( 𝜑 → 𝑇 ∈ ( SubRing ‘ 𝑅 ) )
7 reldmpsr ⊢ Rel dom mPwSer
8 7 3 4 elbasov ⊢ ( 𝑋 ∈ 𝐵 → ( 𝐼 ∈ V ∧ 𝐻 ∈ V ) )
9 8 ad2antrl ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐼 ∈ V ∧ 𝐻 ∈ V ) )
10 9 simpld ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐼 ∈ V )
11 eqid ⊢ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
12 11 psrbaglefi ⊢ ( ( 𝐼 ∈ V ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) → { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ∈ Fin )
13 10 12 sylan ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) → { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ∈ Fin )
14 subrgsubg ⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝑇 ∈ ( SubGrp ‘ 𝑅 ) )
15 6 14 syl ⊢ ( 𝜑 → 𝑇 ∈ ( SubGrp ‘ 𝑅 ) )
16 subgsubm ⊢ ( 𝑇 ∈ ( SubGrp ‘ 𝑅 ) → 𝑇 ∈ ( SubMnd ‘ 𝑅 ) )
17 15 16 syl ⊢ ( 𝜑 → 𝑇 ∈ ( SubMnd ‘ 𝑅 ) )
18 17 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) → 𝑇 ∈ ( SubMnd ‘ 𝑅 ) )
19 6 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑇 ∈ ( SubRing ‘ 𝑅 ) )
20 eqid ⊢ ( Base ‘ 𝐻 ) = ( Base ‘ 𝐻 )
21 simprl ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
22 3 20 11 4 21 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝐻 ) )
23 22 adantr ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) → 𝑋 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝐻 ) )
24 elrabi ⊢ ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } → 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } )
25 ffvelrn ⊢ ( ( 𝑋 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝐻 ) ∧ 𝑥 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) → ( 𝑋 ‘ 𝑥 ) ∈ ( Base ‘ 𝐻 ) )
26 23 24 25 syl2an ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑋 ‘ 𝑥 ) ∈ ( Base ‘ 𝐻 ) )
27 2 subrgbas ⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝑇 = ( Base ‘ 𝐻 ) )
28 19 27 syl ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑇 = ( Base ‘ 𝐻 ) )
29 26 28 eleqtrrd ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑋 ‘ 𝑥 ) ∈ 𝑇 )
30 simprr ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
31 3 20 11 4 30 psrelbas ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝐻 ) )
32 31 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑌 : { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ⟶ ( Base ‘ 𝐻 ) )
33 ssrab2 ⊢ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ⊆ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin }
34 10 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝐼 ∈ V )
35 simplr ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } )
36 simpr ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } )
37 eqid ⊢ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } = { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 }
38 11 37 psrbagconcl ⊢ ( ( 𝐼 ∈ V ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑥 ) ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } )
39 34 35 36 38 syl3anc ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑥 ) ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } )
40 33 39 sseldi ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑘 ∘f − 𝑥 ) ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } )
41 32 40 ffvelrnd ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ∈ ( Base ‘ 𝐻 ) )
42 41 28 eleqtrrd ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ∈ 𝑇 )
43 eqid ⊢ ( .r ‘ 𝑅 ) = ( .r ‘ 𝑅 )
44 43 subrgmcl ⊢ ( ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) ∧ ( 𝑋 ‘ 𝑥 ) ∈ 𝑇 ∧ ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ∈ 𝑇 ) → ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ∈ 𝑇 )
45 19 29 42 44 syl3anc ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ∈ 𝑇 )
46 45 fmpttd ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) → ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) : { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ⟶ 𝑇 )
47 13 18 46 2 gsumsubm ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) = ( 𝐻 Σg ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) )
48 2 43 ressmulr ⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → ( .r ‘ 𝑅 ) = ( .r ‘ 𝐻 ) )
49 6 48 syl ⊢ ( 𝜑 → ( .r ‘ 𝑅 ) = ( .r ‘ 𝐻 ) )
50 49 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( .r ‘ 𝑅 ) = ( .r ‘ 𝐻 ) )
51 50 oveqd ⊢ ( ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ∧ 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ) → ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) = ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝐻 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) )
52 51 mpteq2dva ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) → ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) = ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝐻 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) )
53 52 oveq2d ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) → ( 𝐻 Σg ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) = ( 𝐻 Σg ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝐻 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) )
54 47 53 eqtrd ⊢ ( ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) ∧ 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) → ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) = ( 𝐻 Σg ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝐻 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) )
55 54 mpteq2dva ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) = ( 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ ( 𝐻 Σg ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝐻 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )
56 eqid ⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 )
57 eqid ⊢ ( .r ‘ 𝑆 ) = ( .r ‘ 𝑆 )
58 fvex ⊢ ( Base ‘ 𝑅 ) ∈ V
59 6 27 syl ⊢ ( 𝜑 → 𝑇 = ( Base ‘ 𝐻 ) )
60 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
61 60 subrgss ⊢ ( 𝑇 ∈ ( SubRing ‘ 𝑅 ) → 𝑇 ⊆ ( Base ‘ 𝑅 ) )
62 6 61 syl ⊢ ( 𝜑 → 𝑇 ⊆ ( Base ‘ 𝑅 ) )
63 59 62 eqsstrrd ⊢ ( 𝜑 → ( Base ‘ 𝐻 ) ⊆ ( Base ‘ 𝑅 ) )
64 mapss ⊢ ( ( ( Base ‘ 𝑅 ) ∈ V ∧ ( Base ‘ 𝐻 ) ⊆ ( Base ‘ 𝑅 ) ) → ( ( Base ‘ 𝐻 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ⊆ ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) )
65 58 63 64 sylancr ⊢ ( 𝜑 → ( ( Base ‘ 𝐻 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ⊆ ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) )
66 65 adantr ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( Base ‘ 𝐻 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) ⊆ ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) )
67 3 20 11 4 10 psrbas ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐵 = ( ( Base ‘ 𝐻 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) )
68 1 60 11 56 10 psrbas ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( Base ‘ 𝑆 ) = ( ( Base ‘ 𝑅 ) ↑m { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ) )
69 66 67 68 3sstr4d ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐵 ⊆ ( Base ‘ 𝑆 ) )
70 69 21 sseldd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ ( Base ‘ 𝑆 ) )
71 69 30 sseldd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ ( Base ‘ 𝑆 ) )
72 1 56 43 57 11 70 71 psrmulfval ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( .r ‘ 𝑆 ) 𝑌 ) = ( 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ ( 𝑅 Σg ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝑅 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )
73 eqid ⊢ ( .r ‘ 𝐻 ) = ( .r ‘ 𝐻 )
74 eqid ⊢ ( .r ‘ 𝑈 ) = ( .r ‘ 𝑈 )
75 3 4 73 74 11 21 30 psrmulfval ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( .r ‘ 𝑈 ) 𝑌 ) = ( 𝑘 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ↦ ( 𝐻 Σg ( 𝑥 ∈ { 𝑦 ∈ { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } ∣ 𝑦 ∘r ≤ 𝑘 } ↦ ( ( 𝑋 ‘ 𝑥 ) ( .r ‘ 𝐻 ) ( 𝑌 ‘ ( 𝑘 ∘f − 𝑥 ) ) ) ) ) ) )
76 55 72 75 3eqtr4rd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( .r ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( .r ‘ 𝑆 ) 𝑌 ) )
77 4 fvexi ⊢ 𝐵 ∈ V
78 5 57 ressmulr ⊢ ( 𝐵 ∈ V → ( .r ‘ 𝑆 ) = ( .r ‘ 𝑃 ) )
79 77 78 mp1i ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( .r ‘ 𝑆 ) = ( .r ‘ 𝑃 ) )
80 79 oveqd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( .r ‘ 𝑆 ) 𝑌 ) = ( 𝑋 ( .r ‘ 𝑃 ) 𝑌 ) )
81 76 80 eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ( .r ‘ 𝑈 ) 𝑌 ) = ( 𝑋 ( .r ‘ 𝑃 ) 𝑌 ) )