Theorem tdeglem3

Description: Additivity of the total degree helper function. (Contributed by Stefan O'Rear, 26-Mar-2015) (Proof shortened by AV, 27-Jul-2019) Remove a sethood antecedent. (Revised by SN, 7-Aug-2024)

	Ref	Expression
Hypotheses	tdeglem.a	⊢ 𝐴 = { 𝑚 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑚 “ ℕ ) ∈ Fin }
	tdeglem.h	⊢ 𝐻 = ( ℎ ∈ 𝐴 ↦ ( ℂfld Σg ℎ ) )
Assertion	tdeglem3	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝐻 ‘ ( 𝑋 ∘f + 𝑌 ) ) = ( ( 𝐻 ‘ 𝑋 ) + ( 𝐻 ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		tdeglem.a	⊢ 𝐴 = { 𝑚 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑚 “ ℕ ) ∈ Fin }
2		tdeglem.h	⊢ 𝐻 = ( ℎ ∈ 𝐴 ↦ ( ℂfld Σg ℎ ) )
3		cnfldbas	⊢ ℂ = ( Base ‘ ℂfld )
4		cnfld0	⊢ 0 = ( 0g ‘ ℂfld )
5		cnfldadd	⊢ + = ( +g ‘ ℂfld )
6		cnring	⊢ ℂfld ∈ Ring
7		ringcmn	⊢ ( ℂfld ∈ Ring → ℂfld ∈ CMnd )
8	6 7	mp1i	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ℂfld ∈ CMnd )
9		simpl	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 ∈ 𝐴 )
10	1	psrbagf	⊢ ( 𝑋 ∈ 𝐴 → 𝑋 : 𝐼 ⟶ ℕ0 )
11		nn0sscn	⊢ ℕ0 ⊆ ℂ
12		fss	⊢ ( ( 𝑋 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑋 : 𝐼 ⟶ ℂ )
13	10 11 12	sylancl	⊢ ( 𝑋 ∈ 𝐴 → 𝑋 : 𝐼 ⟶ ℂ )
14	13	adantr	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 : 𝐼 ⟶ ℂ )
15	14	ffnd	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 Fn 𝐼 )
16	9 15	fndmexd	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝐼 ∈ V )
17	1	psrbagf	⊢ ( 𝑌 ∈ 𝐴 → 𝑌 : 𝐼 ⟶ ℕ0 )
18		fss	⊢ ( ( 𝑌 : 𝐼 ⟶ ℕ0 ∧ ℕ0 ⊆ ℂ ) → 𝑌 : 𝐼 ⟶ ℂ )
19	17 11 18	sylancl	⊢ ( 𝑌 ∈ 𝐴 → 𝑌 : 𝐼 ⟶ ℂ )
20	19	adantl	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑌 : 𝐼 ⟶ ℂ )
21	1	psrbagfsupp	⊢ ( 𝑋 ∈ 𝐴 → 𝑋 finSupp 0 )
22	21	adantr	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑋 finSupp 0 )
23	1	psrbagfsupp	⊢ ( 𝑌 ∈ 𝐴 → 𝑌 finSupp 0 )
24	23	adantl	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → 𝑌 finSupp 0 )
25	3 4 5 8 16 14 20 22 24	gsumadd	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ℂfld Σg ( 𝑋 ∘f + 𝑌 ) ) = ( ( ℂfld Σg 𝑋 ) + ( ℂfld Σg 𝑌 ) ) )
26	1	psrbagaddcl	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 ∘f + 𝑌 ) ∈ 𝐴 )
27		oveq2	⊢ ( ℎ = ( 𝑋 ∘f + 𝑌 ) → ( ℂfld Σg ℎ ) = ( ℂfld Σg ( 𝑋 ∘f + 𝑌 ) ) )
28		ovex	⊢ ( ℂfld Σg ( 𝑋 ∘f + 𝑌 ) ) ∈ V
29	27 2 28	fvmpt	⊢ ( ( 𝑋 ∘f + 𝑌 ) ∈ 𝐴 → ( 𝐻 ‘ ( 𝑋 ∘f + 𝑌 ) ) = ( ℂfld Σg ( 𝑋 ∘f + 𝑌 ) ) )
30	26 29	syl	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝐻 ‘ ( 𝑋 ∘f + 𝑌 ) ) = ( ℂfld Σg ( 𝑋 ∘f + 𝑌 ) ) )
31		oveq2	⊢ ( ℎ = 𝑋 → ( ℂfld Σg ℎ ) = ( ℂfld Σg 𝑋 ) )
32		ovex	⊢ ( ℂfld Σg 𝑋 ) ∈ V
33	31 2 32	fvmpt	⊢ ( 𝑋 ∈ 𝐴 → ( 𝐻 ‘ 𝑋 ) = ( ℂfld Σg 𝑋 ) )
34		oveq2	⊢ ( ℎ = 𝑌 → ( ℂfld Σg ℎ ) = ( ℂfld Σg 𝑌 ) )
35		ovex	⊢ ( ℂfld Σg 𝑌 ) ∈ V
36	34 2 35	fvmpt	⊢ ( 𝑌 ∈ 𝐴 → ( 𝐻 ‘ 𝑌 ) = ( ℂfld Σg 𝑌 ) )
37	33 36	oveqan12d	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( ( 𝐻 ‘ 𝑋 ) + ( 𝐻 ‘ 𝑌 ) ) = ( ( ℂfld Σg 𝑋 ) + ( ℂfld Σg 𝑌 ) ) )
38	25 30 37	3eqtr4d	⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝐻 ‘ ( 𝑋 ∘f + 𝑌 ) ) = ( ( 𝐻 ‘ 𝑋 ) + ( 𝐻 ‘ 𝑌 ) ) )