Metamath Proof Explorer

Theorem mplvrpmmhm

Description: The action of permuting variables in a multivariate polynomial is a monoid homomorphism. (Contributed by Thierry Arnoux, 11-Jan-2026)

Ref Expression
Hypotheses mplvrpmga.1 ⊢ 𝑆 = ( SymGrp ‘ 𝐼 )
mplvrpmga.2 ⊢ 𝑃 = ( Base ‘ 𝑆 )
mplvrpmga.3 ⊢ 𝑀 = ( Base ‘ ( 𝐼 mPoly 𝑅 ) )
mplvrpmga.4 ⊢ 𝐴 = ( 𝑑 ∈ 𝑃 , 𝑓 ∈ 𝑀 ↦ ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) )
mplvrpmga.5 ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
mplvrpmmhm.f ⊢ 𝐹 = ( 𝑓 ∈ 𝑀 ↦ ( 𝐷 𝐴 𝑓 ) )
mplvrpmmhm.w ⊢ 𝑊 = ( 𝐼 mPoly 𝑅 )
mplvrpmmhm.1 ⊢ ( 𝜑 → 𝑅 ∈ Ring )
mplvrpmmhm.2 ⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
Assertion mplvrpmmhm ( 𝜑 → 𝐹 ∈ ( 𝑊 MndHom 𝑊 ) )

Proof

Step Hyp Ref Expression
1 mplvrpmga.1 ⊢ 𝑆 = ( SymGrp ‘ 𝐼 )
2 mplvrpmga.2 ⊢ 𝑃 = ( Base ‘ 𝑆 )
3 mplvrpmga.3 ⊢ 𝑀 = ( Base ‘ ( 𝐼 mPoly 𝑅 ) )
4 mplvrpmga.4 ⊢ 𝐴 = ( 𝑑 ∈ 𝑃 , 𝑓 ∈ 𝑀 ↦ ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) )
5 mplvrpmga.5 ⊢ ( 𝜑 → 𝐼 ∈ 𝑉 )
6 mplvrpmmhm.f ⊢ 𝐹 = ( 𝑓 ∈ 𝑀 ↦ ( 𝐷 𝐴 𝑓 ) )
7 mplvrpmmhm.w ⊢ 𝑊 = ( 𝐼 mPoly 𝑅 )
8 mplvrpmmhm.1 ⊢ ( 𝜑 → 𝑅 ∈ Ring )
9 mplvrpmmhm.2 ⊢ ( 𝜑 → 𝐷 ∈ 𝑃 )
10 7 fveq2i ⊢ ( Base ‘ 𝑊 ) = ( Base ‘ ( 𝐼 mPoly 𝑅 ) )
11 3 10 eqtr4i ⊢ 𝑀 = ( Base ‘ 𝑊 )
12 eqid ⊢ ( +g ‘ 𝑊 ) = ( +g ‘ 𝑊 )
13 eqid ⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 )
14 7 5 8 mplringd ⊢ ( 𝜑 → 𝑊 ∈ Ring )
15 14 ringgrpd ⊢ ( 𝜑 → 𝑊 ∈ Grp )
16 15 grpmndd ⊢ ( 𝜑 → 𝑊 ∈ Mnd )
17 1 2 3 4 5 mplvrpmga ⊢ ( 𝜑 → 𝐴 ∈ ( 𝑆 GrpAct 𝑀 ) )
18 2 gaf ⊢ ( 𝐴 ∈ ( 𝑆 GrpAct 𝑀 ) → 𝐴 : ( 𝑃 × 𝑀 ) ⟶ 𝑀 )
19 17 18 syl ⊢ ( 𝜑 → 𝐴 : ( 𝑃 × 𝑀 ) ⟶ 𝑀 )
20 19 fovcld ⊢ ( ( 𝜑 ∧ 𝐷 ∈ 𝑃 ∧ 𝑓 ∈ 𝑀 ) → ( 𝐷 𝐴 𝑓 ) ∈ 𝑀 )
21 20 3expa ⊢ ( ( ( 𝜑 ∧ 𝐷 ∈ 𝑃 ) ∧ 𝑓 ∈ 𝑀 ) → ( 𝐷 𝐴 𝑓 ) ∈ 𝑀 )
22 21 an32s ⊢ ( ( ( 𝜑 ∧ 𝑓 ∈ 𝑀 ) ∧ 𝐷 ∈ 𝑃 ) → ( 𝐷 𝐴 𝑓 ) ∈ 𝑀 )
23 9 22 mpidan ⊢ ( ( 𝜑 ∧ 𝑓 ∈ 𝑀 ) → ( 𝐷 𝐴 𝑓 ) ∈ 𝑀 )
24 23 6 fmptd ⊢ ( 𝜑 → 𝐹 : 𝑀 ⟶ 𝑀 )
25 eqid ⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 )
26 eqid ⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 }
27 26 psrbasfsupp ⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } = { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ ℎ “ ℕ ) ∈ Fin }
28 simplr ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝑖 ∈ 𝑀 )
29 7 25 11 27 28 mplelf ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝑖 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ⟶ ( Base ‘ 𝑅 ) )
30 29 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑖 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ⟶ ( Base ‘ 𝑅 ) )
31 30 ffnd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑖 Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
32 simpr ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝑗 ∈ 𝑀 )
33 7 25 11 27 32 mplelf ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝑗 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ⟶ ( Base ‘ 𝑅 ) )
34 33 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑗 : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ⟶ ( Base ‘ 𝑅 ) )
35 34 ffnd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑗 Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
36 ovex ⊢ ( ℕ0 ↑m 𝐼 ) ∈ V
37 36 rabex ⊢ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ∈ V
38 37 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ∈ V )
39 breq1 ⊢ ( ℎ = ( 𝑥 ∘ 𝐷 ) → ( ℎ finSupp 0 ↔ ( 𝑥 ∘ 𝐷 ) finSupp 0 ) )
40 nn0ex ⊢ ℕ0 ∈ V
41 40 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ℕ0 ∈ V )
42 5 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐼 ∈ 𝑉 )
43 5 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐼 ∈ 𝑉 )
44 40 a1i ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ℕ0 ∈ V )
45 breq1 ⊢ ( ℎ = 𝑥 → ( ℎ finSupp 0 ↔ 𝑥 finSupp 0 ) )
46 45 elrab ⊢ ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↔ ( 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∧ 𝑥 finSupp 0 ) )
47 46 bilani ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∧ 𝑥 finSupp 0 ) )
48 47 simpld ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) )
49 43 44 48 elmaprd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑥 : 𝐼 ⟶ ℕ0 )
50 49 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑥 : 𝐼 ⟶ ℕ0 )
51 1 2 symgbasf1o ⊢ ( 𝐷 ∈ 𝑃 → 𝐷 : 𝐼 –1-1-onto→ 𝐼 )
52 9 51 syl ⊢ ( 𝜑 → 𝐷 : 𝐼 –1-1-onto→ 𝐼 )
53 f1of ⊢ ( 𝐷 : 𝐼 –1-1-onto→ 𝐼 → 𝐷 : 𝐼 ⟶ 𝐼 )
54 52 53 syl ⊢ ( 𝜑 → 𝐷 : 𝐼 ⟶ 𝐼 )
55 54 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐷 : 𝐼 ⟶ 𝐼 )
56 50 55 fcod ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) : 𝐼 ⟶ ℕ0 )
57 41 42 56 elmapdd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) ∈ ( ℕ0 ↑m 𝐼 ) )
58 47 simprd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑥 finSupp 0 )
59 52 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐷 : 𝐼 –1-1-onto→ 𝐼 )
60 f1of1 ⊢ ( 𝐷 : 𝐼 –1-1-onto→ 𝐼 → 𝐷 : 𝐼 –1-1→ 𝐼 )
61 59 60 syl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐷 : 𝐼 –1-1→ 𝐼 )
62 0nn0 ⊢ 0 ∈ ℕ0
63 62 a1i ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 0 ∈ ℕ0 )
64 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
65 58 61 63 64 fsuppco ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) finSupp 0 )
66 65 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) finSupp 0 )
67 39 57 66 elrabd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
68 fnfvof ⊢ ( ( ( 𝑖 Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ∧ 𝑗 Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) ∧ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ∈ V ∧ ( 𝑥 ∘ 𝐷 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) ) → ( ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) = ( ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ( +g ‘ 𝑅 ) ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
69 31 35 38 67 68 syl22anc ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) = ( ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ( +g ‘ 𝑅 ) ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
70 oveq2 ⊢ ( 𝑓 = 𝑖 → ( 𝐷 𝐴 𝑓 ) = ( 𝐷 𝐴 𝑖 ) )
71 4 a1i ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝐴 = ( 𝑑 ∈ 𝑃 , 𝑓 ∈ 𝑀 ↦ ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) ) )
72 simpr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑖 ) → 𝑓 = 𝑖 )
73 coeq2 ⊢ ( 𝑑 = 𝐷 → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
74 73 adantr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑖 ) → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
75 72 74 fveq12d ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑖 ) → ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) = ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) )
76 75 mpteq2dv ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑖 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
77 76 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑖 ) ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
78 9 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝐷 ∈ 𝑃 )
79 37 a1i ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ∈ V )
80 79 mptexd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ) ∈ V )
81 71 77 78 28 80 ovmpod ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐷 𝐴 𝑖 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
82 70 81 sylan9eqr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑓 = 𝑖 ) → ( 𝐷 𝐴 𝑓 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
83 6 82 28 80 fvmptd2 ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑖 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
84 fvexd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ∈ V )
85 83 84 fvmpt2d ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) = ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) )
86 oveq2 ⊢ ( 𝑓 = 𝑗 → ( 𝐷 𝐴 𝑓 ) = ( 𝐷 𝐴 𝑗 ) )
87 simpr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑗 ) → 𝑓 = 𝑗 )
88 73 adantr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑗 ) → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
89 87 88 fveq12d ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑗 ) → ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) = ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) )
90 89 mpteq2dv ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑗 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
91 90 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑗 ) ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
92 79 mptexd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) ∈ V )
93 71 91 78 32 92 ovmpod ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐷 𝐴 𝑗 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
94 86 93 sylan9eqr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑓 = 𝑗 ) → ( 𝐷 𝐴 𝑓 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
95 6 94 32 92 fvmptd2 ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
96 fvexd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ∈ V )
97 95 96 fvmpt2d ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑥 ) = ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) )
98 85 97 oveq12d ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ( +g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑥 ) ) = ( ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ( +g ‘ 𝑅 ) ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
99 69 98 eqtr4d ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) = ( ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ( +g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑥 ) ) )
100 99 mpteq2dva ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ( +g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑥 ) ) ) )
101 24 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝐹 : 𝑀 ⟶ 𝑀 )
102 101 28 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑖 ) ∈ 𝑀 )
103 7 25 11 27 102 mplelf ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑖 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ⟶ ( Base ‘ 𝑅 ) )
104 103 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑖 ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
105 101 32 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑗 ) ∈ 𝑀 )
106 7 25 11 27 105 mplelf ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑗 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ⟶ ( Base ‘ 𝑅 ) )
107 106 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑗 ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
108 79 104 107 offvalfv ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( ( 𝐹 ‘ 𝑖 ) ∘f ( +g ‘ 𝑅 ) ( 𝐹 ‘ 𝑗 ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) ( +g ‘ 𝑅 ) ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑥 ) ) ) )
109 100 108 eqtr4d ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) = ( ( 𝐹 ‘ 𝑖 ) ∘f ( +g ‘ 𝑅 ) ( 𝐹 ‘ 𝑗 ) ) )
110 oveq2 ⊢ ( 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) → ( 𝐷 𝐴 𝑓 ) = ( 𝐷 𝐴 ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) )
111 simpr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) → 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) )
112 73 adantr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
113 111 112 fveq12d ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) → ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) = ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) )
114 113 mpteq2dv ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
115 114 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
116 15 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝑊 ∈ Grp )
117 11 12 116 28 32 grpcld ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ∈ 𝑀 )
118 79 mptexd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) ∈ V )
119 71 115 78 117 118 ovmpod ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐷 𝐴 ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
120 110 119 sylan9eqr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) → ( 𝐷 𝐴 𝑓 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
121 6 120 117 118 fvmptd2 ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
122 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
123 7 11 122 12 28 32 mpladd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) = ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) )
124 123 fveq1d ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) = ( ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) )
125 124 mpteq2dv ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
126 121 125 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
127 7 11 122 12 102 105 mpladd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( ( 𝐹 ‘ 𝑖 ) ( +g ‘ 𝑊 ) ( 𝐹 ‘ 𝑗 ) ) = ( ( 𝐹 ‘ 𝑖 ) ∘f ( +g ‘ 𝑅 ) ( 𝐹 ‘ 𝑗 ) ) )
128 109 126 127 3eqtr4d ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) = ( ( 𝐹 ‘ 𝑖 ) ( +g ‘ 𝑊 ) ( 𝐹 ‘ 𝑗 ) ) )
129 128 anasss ⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑗 ∈ 𝑀 ) ) → ( 𝐹 ‘ ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) = ( ( 𝐹 ‘ 𝑖 ) ( +g ‘ 𝑊 ) ( 𝐹 ‘ 𝑗 ) ) )
130 simpr ⊢ ( ( 𝜑 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) → 𝑓 = ( 0g ‘ 𝑊 ) )
131 130 oveq2d ⊢ ( ( 𝜑 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) → ( 𝐷 𝐴 𝑓 ) = ( 𝐷 𝐴 ( 0g ‘ 𝑊 ) ) )
132 4 a1i ⊢ ( 𝜑 → 𝐴 = ( 𝑑 ∈ 𝑃 , 𝑓 ∈ 𝑀 ↦ ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) ) )
133 simplrr ⊢ ( ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑓 = ( 0g ‘ 𝑊 ) )
134 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
135 8 ringgrpd ⊢ ( 𝜑 → 𝑅 ∈ Grp )
136 7 27 134 13 5 135 mpl0 ⊢ ( 𝜑 → ( 0g ‘ 𝑊 ) = ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) )
137 136 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 0g ‘ 𝑊 ) = ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) )
138 133 137 eqtrd ⊢ ( ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑓 = ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) )
139 73 ad2antrl ⊢ ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
140 139 adantr ⊢ ( ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
141 138 140 fveq12d ⊢ ( ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) = ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) ‘ ( 𝑥 ∘ 𝐷 ) ) )
142 141 mpteq2dva ⊢ ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
143 54 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐷 : 𝐼 ⟶ 𝐼 )
144 49 143 fcod ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) : 𝐼 ⟶ ℕ0 )
145 44 43 144 elmapdd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) ∈ ( ℕ0 ↑m 𝐼 ) )
146 39 145 65 elrabd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
147 fvex ⊢ ( 0g ‘ 𝑅 ) ∈ V
148 147 fvconst2 ⊢ ( ( 𝑥 ∘ 𝐷 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } → ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) ‘ ( 𝑥 ∘ 𝐷 ) ) = ( 0g ‘ 𝑅 ) )
149 146 148 syl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) ‘ ( 𝑥 ∘ 𝐷 ) ) = ( 0g ‘ 𝑅 ) )
150 149 mpteq2dva ⊢ ( 𝜑 → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 0g ‘ 𝑅 ) ) )
151 fconstmpt ⊢ ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 0g ‘ 𝑅 ) )
152 136 151 eqtrdi ⊢ ( 𝜑 → ( 0g ‘ 𝑊 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 0g ‘ 𝑅 ) ) )
153 150 152 eqtr4d ⊢ ( 𝜑 → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) = ( 0g ‘ 𝑊 ) )
154 153 adantr ⊢ ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) = ( 0g ‘ 𝑊 ) )
155 142 154 eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) = ( 0g ‘ 𝑊 ) )
156 11 13 grpidcl ⊢ ( 𝑊 ∈ Grp → ( 0g ‘ 𝑊 ) ∈ 𝑀 )
157 15 156 syl ⊢ ( 𝜑 → ( 0g ‘ 𝑊 ) ∈ 𝑀 )
158 132 155 9 157 157 ovmpod ⊢ ( 𝜑 → ( 𝐷 𝐴 ( 0g ‘ 𝑊 ) ) = ( 0g ‘ 𝑊 ) )
159 158 adantr ⊢ ( ( 𝜑 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) → ( 𝐷 𝐴 ( 0g ‘ 𝑊 ) ) = ( 0g ‘ 𝑊 ) )
160 131 159 eqtrd ⊢ ( ( 𝜑 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) → ( 𝐷 𝐴 𝑓 ) = ( 0g ‘ 𝑊 ) )
161 6 160 157 157 fvmptd2 ⊢ ( 𝜑 → ( 𝐹 ‘ ( 0g ‘ 𝑊 ) ) = ( 0g ‘ 𝑊 ) )
162 11 11 12 12 13 13 16 16 24 129 161 ismhmd ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑊 MndHom 𝑊 ) )