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 breq1 ⊢ ( ℎ = 𝑥 → ( ℎ finSupp 0 ↔ 𝑥 finSupp 0 ) )
44 43 elrab ⊢ ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↔ ( 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∧ 𝑥 finSupp 0 ) )
45 44 bilani ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) ∧ 𝑥 finSupp 0 ) )
46 45 simpld ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑥 ∈ ( ℕ0 ↑m 𝐼 ) )
47 46 elmaprd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑥 : 𝐼 ⟶ ℕ0 )
48 47 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑥 : 𝐼 ⟶ ℕ0 )
49 1 2 symgbasf1o ⊢ ( 𝐷 ∈ 𝑃 → 𝐷 : 𝐼 –1-1-onto→ 𝐼 )
50 9 49 syl ⊢ ( 𝜑 → 𝐷 : 𝐼 –1-1-onto→ 𝐼 )
51 f1of ⊢ ( 𝐷 : 𝐼 –1-1-onto→ 𝐼 → 𝐷 : 𝐼 ⟶ 𝐼 )
52 50 51 syl ⊢ ( 𝜑 → 𝐷 : 𝐼 ⟶ 𝐼 )
53 52 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐷 : 𝐼 ⟶ 𝐼 )
54 48 53 fcod ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) : 𝐼 ⟶ ℕ0 )
55 41 42 54 elmapdd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) ∈ ( ℕ0 ↑m 𝐼 ) )
56 45 simprd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑥 finSupp 0 )
57 50 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐷 : 𝐼 –1-1-onto→ 𝐼 )
58 f1of1 ⊢ ( 𝐷 : 𝐼 –1-1-onto→ 𝐼 → 𝐷 : 𝐼 –1-1→ 𝐼 )
59 57 58 syl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐷 : 𝐼 –1-1→ 𝐼 )
60 0nn0 ⊢ 0 ∈ ℕ0
61 60 a1i ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 0 ∈ ℕ0 )
62 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
63 56 59 61 62 fsuppco ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) finSupp 0 )
64 63 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) finSupp 0 )
65 39 55 64 elrabd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
66 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 ) ) ) )
67 31 35 38 65 66 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 ) ) ) )
68 oveq2 ⊢ ( 𝑓 = 𝑖 → ( 𝐷 𝐴 𝑓 ) = ( 𝐷 𝐴 𝑖 ) )
69 4 a1i ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝐴 = ( 𝑑 ∈ 𝑃 , 𝑓 ∈ 𝑀 ↦ ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) ) )
70 simpr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑖 ) → 𝑓 = 𝑖 )
71 coeq2 ⊢ ( 𝑑 = 𝐷 → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
72 71 adantr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑖 ) → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
73 70 72 fveq12d ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑖 ) → ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) = ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) )
74 73 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 ) ) ) )
75 74 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 ) ) ) )
76 9 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝐷 ∈ 𝑃 )
77 37 a1i ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ∈ V )
78 77 mptexd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ) ∈ V )
79 69 75 76 28 78 ovmpod ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐷 𝐴 𝑖 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
80 68 79 sylan9eqr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑓 = 𝑖 ) → ( 𝐷 𝐴 𝑓 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
81 6 80 28 78 fvmptd2 ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑖 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
82 fvexd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) ∈ V )
83 81 82 fvmpt2d ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( ( 𝐹 ‘ 𝑖 ) ‘ 𝑥 ) = ( 𝑖 ‘ ( 𝑥 ∘ 𝐷 ) ) )
84 oveq2 ⊢ ( 𝑓 = 𝑗 → ( 𝐷 𝐴 𝑓 ) = ( 𝐷 𝐴 𝑗 ) )
85 simpr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑗 ) → 𝑓 = 𝑗 )
86 71 adantr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑗 ) → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
87 85 86 fveq12d ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = 𝑗 ) → ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) = ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) )
88 87 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 ) ) ) )
89 88 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 ) ) ) )
90 77 mptexd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) ∈ V )
91 69 89 76 32 90 ovmpod ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐷 𝐴 𝑗 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
92 84 91 sylan9eqr ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑓 = 𝑗 ) → ( 𝐷 𝐴 𝑓 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
93 6 92 32 90 fvmptd2 ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑗 ) = ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ) )
94 fvexd ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) ∈ V )
95 93 94 fvmpt2d ⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( ( 𝐹 ‘ 𝑗 ) ‘ 𝑥 ) = ( 𝑗 ‘ ( 𝑥 ∘ 𝐷 ) ) )
96 83 95 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 ) ) ) )
97 67 96 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 ) ) )
98 97 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 ) ) ) )
99 24 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝐹 : 𝑀 ⟶ 𝑀 )
100 99 28 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑖 ) ∈ 𝑀 )
101 7 25 11 27 100 mplelf ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑖 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ⟶ ( Base ‘ 𝑅 ) )
102 101 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑖 ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
103 99 32 ffvelcdmd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑗 ) ∈ 𝑀 )
104 7 25 11 27 103 mplelf ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑗 ) : { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ⟶ ( Base ‘ 𝑅 ) )
105 104 ffnd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ 𝑗 ) Fn { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } )
106 77 102 105 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 ) ) ) )
107 98 106 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 ) ) )
108 oveq2 ⊢ ( 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) → ( 𝐷 𝐴 𝑓 ) = ( 𝐷 𝐴 ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) )
109 simpr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) → 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) )
110 71 adantr ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
111 109 110 fveq12d ⊢ ( ( 𝑑 = 𝐷 ∧ 𝑓 = ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) → ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) = ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) )
112 111 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 ) ) ) )
113 112 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 ) ) ) )
114 15 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → 𝑊 ∈ Grp )
115 11 12 114 28 32 grpcld ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ∈ 𝑀 )
116 77 mptexd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) ) ∈ V )
117 69 113 76 115 116 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 ) ) ) )
118 108 117 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 ) ) ) )
119 6 118 115 116 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 ) ) ) )
120 eqid ⊢ ( +g ‘ 𝑅 ) = ( +g ‘ 𝑅 )
121 7 11 120 12 28 32 mpladd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) = ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) )
122 121 fveq1d ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) = ( ( 𝑖 ∘f ( +g ‘ 𝑅 ) 𝑗 ) ‘ ( 𝑥 ∘ 𝐷 ) ) )
123 122 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 ) ) ) )
124 119 123 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 ) ) ) )
125 7 11 120 12 100 103 mpladd ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( ( 𝐹 ‘ 𝑖 ) ( +g ‘ 𝑊 ) ( 𝐹 ‘ 𝑗 ) ) = ( ( 𝐹 ‘ 𝑖 ) ∘f ( +g ‘ 𝑅 ) ( 𝐹 ‘ 𝑗 ) ) )
126 107 124 125 3eqtr4d ⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ 𝑀 ) ∧ 𝑗 ∈ 𝑀 ) → ( 𝐹 ‘ ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) = ( ( 𝐹 ‘ 𝑖 ) ( +g ‘ 𝑊 ) ( 𝐹 ‘ 𝑗 ) ) )
127 126 anasss ⊢ ( ( 𝜑 ∧ ( 𝑖 ∈ 𝑀 ∧ 𝑗 ∈ 𝑀 ) ) → ( 𝐹 ‘ ( 𝑖 ( +g ‘ 𝑊 ) 𝑗 ) ) = ( ( 𝐹 ‘ 𝑖 ) ( +g ‘ 𝑊 ) ( 𝐹 ‘ 𝑗 ) ) )
128 simpr ⊢ ( ( 𝜑 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) → 𝑓 = ( 0g ‘ 𝑊 ) )
129 128 oveq2d ⊢ ( ( 𝜑 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) → ( 𝐷 𝐴 𝑓 ) = ( 𝐷 𝐴 ( 0g ‘ 𝑊 ) ) )
130 4 a1i ⊢ ( 𝜑 → 𝐴 = ( 𝑑 ∈ 𝑃 , 𝑓 ∈ 𝑀 ↦ ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) ) )
131 simplrr ⊢ ( ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝑓 = ( 0g ‘ 𝑊 ) )
132 eqid ⊢ ( 0g ‘ 𝑅 ) = ( 0g ‘ 𝑅 )
133 8 ringgrpd ⊢ ( 𝜑 → 𝑅 ∈ Grp )
134 7 27 132 13 5 133 mpl0 ⊢ ( 𝜑 → ( 0g ‘ 𝑊 ) = ( { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } × { ( 0g ‘ 𝑅 ) } ) )
135 134 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 ) } ) )
136 131 135 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 ) } ) )
137 71 ad2antrl ⊢ ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
138 137 adantr ⊢ ( ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝑑 ) = ( 𝑥 ∘ 𝐷 ) )
139 136 138 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 ) ) )
140 139 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 ) ) ) )
141 40 a1i ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ℕ0 ∈ V )
142 5 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐼 ∈ 𝑉 )
143 52 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → 𝐷 : 𝐼 ⟶ 𝐼 )
144 47 143 fcod ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) : 𝐼 ⟶ ℕ0 )
145 141 142 144 elmapdd ⊢ ( ( 𝜑 ∧ 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ) → ( 𝑥 ∘ 𝐷 ) ∈ ( ℕ0 ↑m 𝐼 ) )
146 39 145 63 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 ) )
152 134 151 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 ) )
155 140 154 eqtrd ⊢ ( ( 𝜑 ∧ ( 𝑑 = 𝐷 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) ) → ( 𝑥 ∈ { ℎ ∈ ( ℕ0 ↑m 𝐼 ) ∣ ℎ finSupp 0 } ↦ ( 𝑓 ‘ ( 𝑥 ∘ 𝑑 ) ) ) = ( 0g ‘ 𝑊 ) )
156 11 13 15 grpidcld ⊢ ( 𝜑 → ( 0g ‘ 𝑊 ) ∈ 𝑀 )
157 130 155 9 156 156 ovmpod ⊢ ( 𝜑 → ( 𝐷 𝐴 ( 0g ‘ 𝑊 ) ) = ( 0g ‘ 𝑊 ) )
158 157 adantr ⊢ ( ( 𝜑 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) → ( 𝐷 𝐴 ( 0g ‘ 𝑊 ) ) = ( 0g ‘ 𝑊 ) )
159 129 158 eqtrd ⊢ ( ( 𝜑 ∧ 𝑓 = ( 0g ‘ 𝑊 ) ) → ( 𝐷 𝐴 𝑓 ) = ( 0g ‘ 𝑊 ) )
160 6 159 156 156 fvmptd2 ⊢ ( 𝜑 → ( 𝐹 ‘ ( 0g ‘ 𝑊 ) ) = ( 0g ‘ 𝑊 ) )
161 11 11 12 12 13 13 16 16 24 127 160 ismhmd ⊢ ( 𝜑 → 𝐹 ∈ ( 𝑊 MndHom 𝑊 ) )