Metamath Proof Explorer


Theorem psrass1lem

Description: A group sum commutation used by psrass1 . (Contributed by Mario Carneiro, 5-Jan-2015) Remove a sethood hypothesis. (Revised by SN, 7-Aug-2024)

Ref Expression
Hypotheses gsumbagdiag.d 𝐷 = { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin }
gsumbagdiag.s 𝑆 = { 𝑦𝐷𝑦r𝐹 }
gsumbagdiag.f ( 𝜑𝐹𝐷 )
gsumbagdiag.b 𝐵 = ( Base ‘ 𝐺 )
gsumbagdiag.g ( 𝜑𝐺 ∈ CMnd )
gsumbagdiag.x ( ( 𝜑 ∧ ( 𝑗𝑆𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ) → 𝑋𝐵 )
psrass1lem.y ( 𝑘 = ( 𝑛f𝑗 ) → 𝑋 = 𝑌 )
Assertion psrass1lem ( 𝜑 → ( 𝐺 Σg ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ) = ( 𝐺 Σg ( 𝑗𝑆 ↦ ( 𝐺 Σg ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 gsumbagdiag.d 𝐷 = { 𝑓 ∈ ( ℕ0m 𝐼 ) ∣ ( 𝑓 “ ℕ ) ∈ Fin }
2 gsumbagdiag.s 𝑆 = { 𝑦𝐷𝑦r𝐹 }
3 gsumbagdiag.f ( 𝜑𝐹𝐷 )
4 gsumbagdiag.b 𝐵 = ( Base ‘ 𝐺 )
5 gsumbagdiag.g ( 𝜑𝐺 ∈ CMnd )
6 gsumbagdiag.x ( ( 𝜑 ∧ ( 𝑗𝑆𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ) → 𝑋𝐵 )
7 psrass1lem.y ( 𝑘 = ( 𝑛f𝑗 ) → 𝑋 = 𝑌 )
8 1 2 3 gsumbagdiaglem ( ( 𝜑 ∧ ( 𝑚𝑆𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ) ) → ( 𝑗𝑆𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) )
9 6 anassrs ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝑋𝐵 )
10 9 fmpttd ( ( 𝜑𝑗𝑆 ) → ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ⟶ 𝐵 )
11 2 ssrab3 𝑆𝐷
12 1 2 psrbagconcl ( ( 𝐹𝐷𝑗𝑆 ) → ( 𝐹f𝑗 ) ∈ 𝑆 )
13 3 12 sylan ( ( 𝜑𝑗𝑆 ) → ( 𝐹f𝑗 ) ∈ 𝑆 )
14 11 13 sselid ( ( 𝜑𝑗𝑆 ) → ( 𝐹f𝑗 ) ∈ 𝐷 )
15 eqid { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } = { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) }
16 1 15 psrbagconf1o ( ( 𝐹f𝑗 ) ∈ 𝐷 → ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } –1-1-onto→ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } )
17 14 16 syl ( ( 𝜑𝑗𝑆 ) → ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } –1-1-onto→ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } )
18 f1of ( ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } –1-1-onto→ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } → ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ⟶ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } )
19 17 18 syl ( ( 𝜑𝑗𝑆 ) → ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ⟶ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } )
20 10 19 fcod ( ( 𝜑𝑗𝑆 ) → ( ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ∘ ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ⟶ 𝐵 )
21 3 adantr ( ( 𝜑𝑗𝑆 ) → 𝐹𝐷 )
22 21 adantr ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝐹𝐷 )
23 1 psrbagf ( 𝐹𝐷𝐹 : 𝐼 ⟶ ℕ0 )
24 22 23 syl ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝐹 : 𝐼 ⟶ ℕ0 )
25 24 ffvelrnda ( ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ∧ 𝑧𝐼 ) → ( 𝐹𝑧 ) ∈ ℕ0 )
26 simplr ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝑗𝑆 )
27 11 26 sselid ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝑗𝐷 )
28 1 psrbagf ( 𝑗𝐷𝑗 : 𝐼 ⟶ ℕ0 )
29 27 28 syl ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝑗 : 𝐼 ⟶ ℕ0 )
30 29 ffvelrnda ( ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ∧ 𝑧𝐼 ) → ( 𝑗𝑧 ) ∈ ℕ0 )
31 ssrab2 { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ⊆ 𝐷
32 simpr ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } )
33 31 32 sselid ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝑚𝐷 )
34 1 psrbagf ( 𝑚𝐷𝑚 : 𝐼 ⟶ ℕ0 )
35 33 34 syl ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝑚 : 𝐼 ⟶ ℕ0 )
36 35 ffvelrnda ( ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ∧ 𝑧𝐼 ) → ( 𝑚𝑧 ) ∈ ℕ0 )
37 nn0cn ( ( 𝐹𝑧 ) ∈ ℕ0 → ( 𝐹𝑧 ) ∈ ℂ )
38 nn0cn ( ( 𝑗𝑧 ) ∈ ℕ0 → ( 𝑗𝑧 ) ∈ ℂ )
39 nn0cn ( ( 𝑚𝑧 ) ∈ ℕ0 → ( 𝑚𝑧 ) ∈ ℂ )
40 sub32 ( ( ( 𝐹𝑧 ) ∈ ℂ ∧ ( 𝑗𝑧 ) ∈ ℂ ∧ ( 𝑚𝑧 ) ∈ ℂ ) → ( ( ( 𝐹𝑧 ) − ( 𝑗𝑧 ) ) − ( 𝑚𝑧 ) ) = ( ( ( 𝐹𝑧 ) − ( 𝑚𝑧 ) ) − ( 𝑗𝑧 ) ) )
41 37 38 39 40 syl3an ( ( ( 𝐹𝑧 ) ∈ ℕ0 ∧ ( 𝑗𝑧 ) ∈ ℕ0 ∧ ( 𝑚𝑧 ) ∈ ℕ0 ) → ( ( ( 𝐹𝑧 ) − ( 𝑗𝑧 ) ) − ( 𝑚𝑧 ) ) = ( ( ( 𝐹𝑧 ) − ( 𝑚𝑧 ) ) − ( 𝑗𝑧 ) ) )
42 25 30 36 41 syl3anc ( ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ∧ 𝑧𝐼 ) → ( ( ( 𝐹𝑧 ) − ( 𝑗𝑧 ) ) − ( 𝑚𝑧 ) ) = ( ( ( 𝐹𝑧 ) − ( 𝑚𝑧 ) ) − ( 𝑗𝑧 ) ) )
43 42 mpteq2dva ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → ( 𝑧𝐼 ↦ ( ( ( 𝐹𝑧 ) − ( 𝑗𝑧 ) ) − ( 𝑚𝑧 ) ) ) = ( 𝑧𝐼 ↦ ( ( ( 𝐹𝑧 ) − ( 𝑚𝑧 ) ) − ( 𝑗𝑧 ) ) ) )
44 35 ffnd ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝑚 Fn 𝐼 )
45 32 44 fndmexd ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝐼 ∈ V )
46 ovexd ( ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ∧ 𝑧𝐼 ) → ( ( 𝐹𝑧 ) − ( 𝑗𝑧 ) ) ∈ V )
47 24 feqmptd ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝐹 = ( 𝑧𝐼 ↦ ( 𝐹𝑧 ) ) )
48 29 feqmptd ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝑗 = ( 𝑧𝐼 ↦ ( 𝑗𝑧 ) ) )
49 45 25 30 47 48 offval2 ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → ( 𝐹f𝑗 ) = ( 𝑧𝐼 ↦ ( ( 𝐹𝑧 ) − ( 𝑗𝑧 ) ) ) )
50 35 feqmptd ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → 𝑚 = ( 𝑧𝐼 ↦ ( 𝑚𝑧 ) ) )
51 45 46 36 49 50 offval2 ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → ( ( 𝐹f𝑗 ) ∘f𝑚 ) = ( 𝑧𝐼 ↦ ( ( ( 𝐹𝑧 ) − ( 𝑗𝑧 ) ) − ( 𝑚𝑧 ) ) ) )
52 ovexd ( ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ∧ 𝑧𝐼 ) → ( ( 𝐹𝑧 ) − ( 𝑚𝑧 ) ) ∈ V )
53 45 25 36 47 50 offval2 ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → ( 𝐹f𝑚 ) = ( 𝑧𝐼 ↦ ( ( 𝐹𝑧 ) − ( 𝑚𝑧 ) ) ) )
54 45 52 30 53 48 offval2 ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → ( ( 𝐹f𝑚 ) ∘f𝑗 ) = ( 𝑧𝐼 ↦ ( ( ( 𝐹𝑧 ) − ( 𝑚𝑧 ) ) − ( 𝑗𝑧 ) ) ) )
55 43 51 54 3eqtr4d ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → ( ( 𝐹f𝑗 ) ∘f𝑚 ) = ( ( 𝐹f𝑚 ) ∘f𝑗 ) )
56 1 15 psrbagconcl ( ( ( 𝐹f𝑗 ) ∈ 𝐷𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → ( ( 𝐹f𝑗 ) ∘f𝑚 ) ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } )
57 14 56 sylan ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → ( ( 𝐹f𝑗 ) ∘f𝑚 ) ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } )
58 55 57 eqeltrrd ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → ( ( 𝐹f𝑚 ) ∘f𝑗 ) ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } )
59 55 mpteq2dva ( ( 𝜑𝑗𝑆 ) → ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) = ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) ) )
60 nfcv 𝑛 𝑋
61 nfcsb1v 𝑘 𝑛 / 𝑘 𝑋
62 csbeq1a ( 𝑘 = 𝑛𝑋 = 𝑛 / 𝑘 𝑋 )
63 60 61 62 cbvmpt ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) = ( 𝑛 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑛 / 𝑘 𝑋 )
64 63 a1i ( ( 𝜑𝑗𝑆 ) → ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) = ( 𝑛 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑛 / 𝑘 𝑋 ) )
65 csbeq1 ( 𝑛 = ( ( 𝐹f𝑚 ) ∘f𝑗 ) → 𝑛 / 𝑘 𝑋 = ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 )
66 58 59 64 65 fmptco ( ( 𝜑𝑗𝑆 ) → ( ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ∘ ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) ) = ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) )
67 66 feq1d ( ( 𝜑𝑗𝑆 ) → ( ( ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ∘ ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ⟶ 𝐵 ↔ ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ⟶ 𝐵 ) )
68 20 67 mpbid ( ( 𝜑𝑗𝑆 ) → ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ⟶ 𝐵 )
69 68 fvmptelrn ( ( ( 𝜑𝑗𝑆 ) ∧ 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) → ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋𝐵 )
70 69 anasss ( ( 𝜑 ∧ ( 𝑗𝑆𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ) → ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋𝐵 )
71 8 70 syldan ( ( 𝜑 ∧ ( 𝑚𝑆𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ) ) → ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋𝐵 )
72 1 2 3 4 5 71 gsumbagdiag ( 𝜑 → ( 𝐺 Σg ( 𝑚𝑆 , 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) = ( 𝐺 Σg ( 𝑗𝑆 , 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) )
73 eqid ( 0g𝐺 ) = ( 0g𝐺 )
74 1 psrbaglefi ( 𝐹𝐷 → { 𝑦𝐷𝑦r𝐹 } ∈ Fin )
75 3 74 syl ( 𝜑 → { 𝑦𝐷𝑦r𝐹 } ∈ Fin )
76 2 75 eqeltrid ( 𝜑𝑆 ∈ Fin )
77 1 2 psrbagconcl ( ( 𝐹𝐷𝑚𝑆 ) → ( 𝐹f𝑚 ) ∈ 𝑆 )
78 3 77 sylan ( ( 𝜑𝑚𝑆 ) → ( 𝐹f𝑚 ) ∈ 𝑆 )
79 11 78 sselid ( ( 𝜑𝑚𝑆 ) → ( 𝐹f𝑚 ) ∈ 𝐷 )
80 1 psrbaglefi ( ( 𝐹f𝑚 ) ∈ 𝐷 → { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ∈ Fin )
81 79 80 syl ( ( 𝜑𝑚𝑆 ) → { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ∈ Fin )
82 xpfi ( ( 𝑆 ∈ Fin ∧ 𝑆 ∈ Fin ) → ( 𝑆 × 𝑆 ) ∈ Fin )
83 76 76 82 syl2anc ( 𝜑 → ( 𝑆 × 𝑆 ) ∈ Fin )
84 simprl ( ( 𝜑 ∧ ( 𝑚𝑆𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ) ) → 𝑚𝑆 )
85 8 simpld ( ( 𝜑 ∧ ( 𝑚𝑆𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ) ) → 𝑗𝑆 )
86 brxp ( 𝑚 ( 𝑆 × 𝑆 ) 𝑗 ↔ ( 𝑚𝑆𝑗𝑆 ) )
87 84 85 86 sylanbrc ( ( 𝜑 ∧ ( 𝑚𝑆𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ) ) → 𝑚 ( 𝑆 × 𝑆 ) 𝑗 )
88 87 pm2.24d ( ( 𝜑 ∧ ( 𝑚𝑆𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ) ) → ( ¬ 𝑚 ( 𝑆 × 𝑆 ) 𝑗 ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 = ( 0g𝐺 ) ) )
89 88 impr ( ( 𝜑 ∧ ( ( 𝑚𝑆𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ) ∧ ¬ 𝑚 ( 𝑆 × 𝑆 ) 𝑗 ) ) → ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 = ( 0g𝐺 ) )
90 4 73 5 76 81 71 83 89 gsum2d2 ( 𝜑 → ( 𝐺 Σg ( 𝑚𝑆 , 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) = ( 𝐺 Σg ( 𝑚𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ) )
91 1 psrbaglefi ( ( 𝐹f𝑗 ) ∈ 𝐷 → { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ∈ Fin )
92 14 91 syl ( ( 𝜑𝑗𝑆 ) → { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ∈ Fin )
93 simprl ( ( 𝜑 ∧ ( 𝑗𝑆𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ) → 𝑗𝑆 )
94 1 2 3 gsumbagdiaglem ( ( 𝜑 ∧ ( 𝑗𝑆𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ) → ( 𝑚𝑆𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ) )
95 94 simpld ( ( 𝜑 ∧ ( 𝑗𝑆𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ) → 𝑚𝑆 )
96 brxp ( 𝑗 ( 𝑆 × 𝑆 ) 𝑚 ↔ ( 𝑗𝑆𝑚𝑆 ) )
97 93 95 96 sylanbrc ( ( 𝜑 ∧ ( 𝑗𝑆𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ) → 𝑗 ( 𝑆 × 𝑆 ) 𝑚 )
98 97 pm2.24d ( ( 𝜑 ∧ ( 𝑗𝑆𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ) → ( ¬ 𝑗 ( 𝑆 × 𝑆 ) 𝑚 ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 = ( 0g𝐺 ) ) )
99 98 impr ( ( 𝜑 ∧ ( ( 𝑗𝑆𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ∧ ¬ 𝑗 ( 𝑆 × 𝑆 ) 𝑚 ) ) → ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 = ( 0g𝐺 ) )
100 4 73 5 76 92 70 83 99 gsum2d2 ( 𝜑 → ( 𝐺 Σg ( 𝑗𝑆 , 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) = ( 𝐺 Σg ( 𝑗𝑆 ↦ ( 𝐺 Σg ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ) )
101 72 90 100 3eqtr3d ( 𝜑 → ( 𝐺 Σg ( 𝑚𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ) = ( 𝐺 Σg ( 𝑗𝑆 ↦ ( 𝐺 Σg ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ) )
102 5 adantr ( ( 𝜑𝑚𝑆 ) → 𝐺 ∈ CMnd )
103 71 anassrs ( ( ( 𝜑𝑚𝑆 ) ∧ 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ) → ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋𝐵 )
104 103 fmpttd ( ( 𝜑𝑚𝑆 ) → ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) : { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ⟶ 𝐵 )
105 ovex ( ℕ0m 𝐼 ) ∈ V
106 1 105 rabex2 𝐷 ∈ V
107 106 a1i ( ( 𝜑𝑚𝑆 ) → 𝐷 ∈ V )
108 rabexg ( 𝐷 ∈ V → { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ∈ V )
109 mptexg ( { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ∈ V → ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ∈ V )
110 107 108 109 3syl ( ( 𝜑𝑚𝑆 ) → ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ∈ V )
111 funmpt Fun ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 )
112 111 a1i ( ( 𝜑𝑚𝑆 ) → Fun ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) )
113 fvexd ( ( 𝜑𝑚𝑆 ) → ( 0g𝐺 ) ∈ V )
114 suppssdm ( ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) supp ( 0g𝐺 ) ) ⊆ dom ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 )
115 eqid ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) = ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 )
116 115 dmmptss dom ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ⊆ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) }
117 114 116 sstri ( ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) supp ( 0g𝐺 ) ) ⊆ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) }
118 117 a1i ( ( 𝜑𝑚𝑆 ) → ( ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) supp ( 0g𝐺 ) ) ⊆ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } )
119 suppssfifsupp ( ( ( ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ∈ V ∧ Fun ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ∧ ( 0g𝐺 ) ∈ V ) ∧ ( { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ∈ Fin ∧ ( ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) supp ( 0g𝐺 ) ) ⊆ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ) ) → ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) finSupp ( 0g𝐺 ) )
120 110 112 113 81 118 119 syl32anc ( ( 𝜑𝑚𝑆 ) → ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) finSupp ( 0g𝐺 ) )
121 4 73 102 81 104 120 gsumcl ( ( 𝜑𝑚𝑆 ) → ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ∈ 𝐵 )
122 121 fmpttd ( 𝜑 → ( 𝑚𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) : 𝑆𝐵 )
123 1 2 psrbagconf1o ( 𝐹𝐷 → ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) : 𝑆1-1-onto𝑆 )
124 3 123 syl ( 𝜑 → ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) : 𝑆1-1-onto𝑆 )
125 f1ocnv ( ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) : 𝑆1-1-onto𝑆 ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) : 𝑆1-1-onto𝑆 )
126 f1of ( ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) : 𝑆1-1-onto𝑆 ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) : 𝑆𝑆 )
127 124 125 126 3syl ( 𝜑 ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) : 𝑆𝑆 )
128 122 127 fcod ( 𝜑 → ( ( 𝑚𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) : 𝑆𝐵 )
129 coass ( ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) = ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) )
130 f1ococnv2 ( ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) : 𝑆1-1-onto𝑆 → ( ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) = ( I ↾ 𝑆 ) )
131 124 130 syl ( 𝜑 → ( ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) = ( I ↾ 𝑆 ) )
132 131 coeq2d ( 𝜑 → ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) ) = ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( I ↾ 𝑆 ) ) )
133 129 132 syl5eq ( 𝜑 → ( ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) = ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( I ↾ 𝑆 ) ) )
134 eqidd ( 𝜑 → ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) = ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) )
135 eqidd ( 𝜑 → ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) = ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) )
136 breq2 ( 𝑛 = ( 𝐹f𝑚 ) → ( 𝑥r𝑛𝑥r ≤ ( 𝐹f𝑚 ) ) )
137 136 rabbidv ( 𝑛 = ( 𝐹f𝑚 ) → { 𝑥𝐷𝑥r𝑛 } = { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } )
138 ovex ( 𝑛f𝑗 ) ∈ V
139 138 7 csbie ( 𝑛f𝑗 ) / 𝑘 𝑋 = 𝑌
140 oveq1 ( 𝑛 = ( 𝐹f𝑚 ) → ( 𝑛f𝑗 ) = ( ( 𝐹f𝑚 ) ∘f𝑗 ) )
141 140 csbeq1d ( 𝑛 = ( 𝐹f𝑚 ) → ( 𝑛f𝑗 ) / 𝑘 𝑋 = ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 )
142 139 141 eqtr3id ( 𝑛 = ( 𝐹f𝑚 ) → 𝑌 = ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 )
143 137 142 mpteq12dv ( 𝑛 = ( 𝐹f𝑚 ) → ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) = ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) )
144 143 oveq2d ( 𝑛 = ( 𝐹f𝑚 ) → ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) = ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) )
145 78 134 135 144 fmptco ( 𝜑 → ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) = ( 𝑚𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) )
146 145 coeq1d ( 𝜑 → ( ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) = ( ( 𝑚𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) )
147 coires1 ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( I ↾ 𝑆 ) ) = ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ↾ 𝑆 )
148 ssid 𝑆𝑆
149 resmpt ( 𝑆𝑆 → ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ↾ 𝑆 ) = ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) )
150 148 149 ax-mp ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ↾ 𝑆 ) = ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) )
151 147 150 eqtri ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( I ↾ 𝑆 ) ) = ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) )
152 151 a1i ( 𝜑 → ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( I ↾ 𝑆 ) ) = ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) )
153 133 146 152 3eqtr3d ( 𝜑 → ( ( 𝑚𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) = ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) )
154 153 feq1d ( 𝜑 → ( ( ( 𝑚𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) : 𝑆𝐵 ↔ ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) : 𝑆𝐵 ) )
155 128 154 mpbid ( 𝜑 → ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) : 𝑆𝐵 )
156 rabexg ( 𝐷 ∈ V → { 𝑦𝐷𝑦r𝐹 } ∈ V )
157 106 156 mp1i ( 𝜑 → { 𝑦𝐷𝑦r𝐹 } ∈ V )
158 2 157 eqeltrid ( 𝜑𝑆 ∈ V )
159 158 mptexd ( 𝜑 → ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∈ V )
160 funmpt Fun ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) )
161 160 a1i ( 𝜑 → Fun ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) )
162 fvexd ( 𝜑 → ( 0g𝐺 ) ∈ V )
163 suppssdm ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) supp ( 0g𝐺 ) ) ⊆ dom ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) )
164 eqid ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) = ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) )
165 164 dmmptss dom ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ⊆ 𝑆
166 163 165 sstri ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) supp ( 0g𝐺 ) ) ⊆ 𝑆
167 166 a1i ( 𝜑 → ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) supp ( 0g𝐺 ) ) ⊆ 𝑆 )
168 suppssfifsupp ( ( ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∈ V ∧ Fun ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∧ ( 0g𝐺 ) ∈ V ) ∧ ( 𝑆 ∈ Fin ∧ ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) supp ( 0g𝐺 ) ) ⊆ 𝑆 ) ) → ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) finSupp ( 0g𝐺 ) )
169 159 161 162 76 167 168 syl32anc ( 𝜑 → ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) finSupp ( 0g𝐺 ) )
170 4 73 5 76 155 169 124 gsumf1o ( 𝜑 → ( 𝐺 Σg ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ) = ( 𝐺 Σg ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) ) )
171 145 oveq2d ( 𝜑 → ( 𝐺 Σg ( ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ∘ ( 𝑚𝑆 ↦ ( 𝐹f𝑚 ) ) ) ) = ( 𝐺 Σg ( 𝑚𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ) )
172 170 171 eqtrd ( 𝜑 → ( 𝐺 Σg ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ) = ( 𝐺 Σg ( 𝑚𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑚 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ) )
173 5 adantr ( ( 𝜑𝑗𝑆 ) → 𝐺 ∈ CMnd )
174 106 a1i ( ( 𝜑𝑗𝑆 ) → 𝐷 ∈ V )
175 rabexg ( 𝐷 ∈ V → { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ∈ V )
176 mptexg ( { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ∈ V → ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ∈ V )
177 174 175 176 3syl ( ( 𝜑𝑗𝑆 ) → ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ∈ V )
178 funmpt Fun ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 )
179 178 a1i ( ( 𝜑𝑗𝑆 ) → Fun ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) )
180 fvexd ( ( 𝜑𝑗𝑆 ) → ( 0g𝐺 ) ∈ V )
181 suppssdm ( ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) supp ( 0g𝐺 ) ) ⊆ dom ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 )
182 eqid ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) = ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 )
183 182 dmmptss dom ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ⊆ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) }
184 181 183 sstri ( ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) supp ( 0g𝐺 ) ) ⊆ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) }
185 184 a1i ( ( 𝜑𝑗𝑆 ) → ( ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) supp ( 0g𝐺 ) ) ⊆ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } )
186 suppssfifsupp ( ( ( ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ∈ V ∧ Fun ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ∧ ( 0g𝐺 ) ∈ V ) ∧ ( { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ∈ Fin ∧ ( ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) supp ( 0g𝐺 ) ) ⊆ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ) ) → ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) finSupp ( 0g𝐺 ) )
187 177 179 180 92 185 186 syl32anc ( ( 𝜑𝑗𝑆 ) → ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) finSupp ( 0g𝐺 ) )
188 4 73 173 92 10 187 17 gsumf1o ( ( 𝜑𝑗𝑆 ) → ( 𝐺 Σg ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ) = ( 𝐺 Σg ( ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ∘ ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) ) ) )
189 66 oveq2d ( ( 𝜑𝑗𝑆 ) → ( 𝐺 Σg ( ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ∘ ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑗 ) ∘f𝑚 ) ) ) ) = ( 𝐺 Σg ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) )
190 188 189 eqtrd ( ( 𝜑𝑗𝑆 ) → ( 𝐺 Σg ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ) = ( 𝐺 Σg ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) )
191 190 mpteq2dva ( 𝜑 → ( 𝑗𝑆 ↦ ( 𝐺 Σg ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ) ) = ( 𝑗𝑆 ↦ ( 𝐺 Σg ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) )
192 191 oveq2d ( 𝜑 → ( 𝐺 Σg ( 𝑗𝑆 ↦ ( 𝐺 Σg ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ) ) ) = ( 𝐺 Σg ( 𝑗𝑆 ↦ ( 𝐺 Σg ( 𝑚 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ ( ( 𝐹f𝑚 ) ∘f𝑗 ) / 𝑘 𝑋 ) ) ) ) )
193 101 172 192 3eqtr4d ( 𝜑 → ( 𝐺 Σg ( 𝑛𝑆 ↦ ( 𝐺 Σg ( 𝑗 ∈ { 𝑥𝐷𝑥r𝑛 } ↦ 𝑌 ) ) ) ) = ( 𝐺 Σg ( 𝑗𝑆 ↦ ( 𝐺 Σg ( 𝑘 ∈ { 𝑥𝐷𝑥r ≤ ( 𝐹f𝑗 ) } ↦ 𝑋 ) ) ) ) )