mhphf - Mathbox for Steven Nguyen

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Theorem mhphf 41830

Description: A homogeneous polynomial defines a homogeneous function. Equivalently, an algebraic form is a homogeneous function. (An algebraic form is the function corresponding to a homogeneous polynomial, which in this case is the (𝑄‘𝑋) which corresponds to 𝑋). (Contributed by SN, 28-Jul-2024.) (Proof shortened by SN, 8-Mar-2025.)

**Hypotheses**
Ref	Expression
mhphf.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
mhphf.h	⊢ 𝐻 = (𝐼 mHomP 𝑈)
mhphf.u	⊢ 𝑈 = (𝑆 ↾_s 𝑅)
mhphf.k	⊢ 𝐾 = (Base‘𝑆)
mhphf.m	⊢ · = (._r‘𝑆)
mhphf.e	⊢ ↑ = (._g‘(mulGrp‘𝑆))
mhphf.i	⊢ (𝜑 → 𝐼 ∈ 𝑉)
mhphf.s	⊢ (𝜑 → 𝑆 ∈ CRing)
mhphf.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
mhphf.l	⊢ (𝜑 → 𝐿 ∈ 𝑅)
mhphf.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
mhphf.x	⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))
mhphf.a	⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))

**Assertion**
Ref	Expression
mhphf	⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘_f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

Proof of Theorem mhphf Dummy variables 𝑏 𝑖 𝑘 𝑗 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		mhphf.i	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ 𝑉)
2	1	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → 𝐼 ∈ 𝑉)
3		mhphf.l	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐿 ∈ 𝑅)
4	3	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → 𝐿 ∈ 𝑅)
5		mhphf.a	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ (𝐾 ↑_m 𝐼))
6		elmapi 8868	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (𝐾 ↑_m 𝐼) → 𝐴:𝐼⟶𝐾)
7	5, 6	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴:𝐼⟶𝐾)
8	7	ffnd 6723	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 Fn 𝐼)
9	8	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → 𝐴 Fn 𝐼)
10		eqidd 2729	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → (𝐴‘𝑖) = (𝐴‘𝑖))
11	2, 4, 9, 10	ofc1 7711	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖) = (𝐿 · (𝐴‘𝑖)))
12	11	oveq2d 7436	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖)) = ((𝑏‘𝑖) ↑ (𝐿 · (𝐴‘𝑖))))
13		mhphf.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ CRing)
14		eqid 2728	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
15	14	crngmgp 20181	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ CRing → (mulGrp‘𝑆) ∈ CMnd)
16	13, 15	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ CMnd)
17	16	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → (mulGrp‘𝑆) ∈ CMnd)
18		elrabi 3676	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} → 𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
19		eqid 2728	. . . . . . . . . . . . . . . 16 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
20	19	psrbagf 21851	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} → 𝑏:𝐼⟶ℕ₀)
21	18, 20	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} → 𝑏:𝐼⟶ℕ₀)
22	21	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → 𝑏:𝐼⟶ℕ₀)
23	22	ffvelcdmda 7094	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → (𝑏‘𝑖) ∈ ℕ₀)
24		mhphf.r	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
25		mhphf.k	. . . . . . . . . . . . . . . 16 ⊢ 𝐾 = (Base‘𝑆)
26	25	subrgss 20511	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾)
27	24, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ⊆ 𝐾)
28	27, 3	sseldd 3981	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ 𝐾)
29	28	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → 𝐿 ∈ 𝐾)
30	7	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → 𝐴:𝐼⟶𝐾)
31	30	ffvelcdmda 7094	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → (𝐴‘𝑖) ∈ 𝐾)
32	14, 25	mgpbas 20080	. . . . . . . . . . . . 13 ⊢ 𝐾 = (Base‘(mulGrp‘𝑆))
33		mhphf.e	. . . . . . . . . . . . 13 ⊢ ↑ = (._g‘(mulGrp‘𝑆))
34		mhphf.m	. . . . . . . . . . . . . 14 ⊢ · = (._r‘𝑆)
35	14, 34	mgpplusg 20078	. . . . . . . . . . . . 13 ⊢ · = (+_g‘(mulGrp‘𝑆))
36	32, 33, 35	mulgnn0di 19780	. . . . . . . . . . . 12 ⊢ (((mulGrp‘𝑆) ∈ CMnd ∧ ((𝑏‘𝑖) ∈ ℕ₀ ∧ 𝐿 ∈ 𝐾 ∧ (𝐴‘𝑖) ∈ 𝐾)) → ((𝑏‘𝑖) ↑ (𝐿 · (𝐴‘𝑖))) = (((𝑏‘𝑖) ↑ 𝐿) · ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))
37	17, 23, 29, 31, 36	syl13anc 1370	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → ((𝑏‘𝑖) ↑ (𝐿 · (𝐴‘𝑖))) = (((𝑏‘𝑖) ↑ 𝐿) · ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))
38	12, 37	eqtrd 2768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖)) = (((𝑏‘𝑖) ↑ 𝐿) · ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))
39	38	mpteq2dva 5248	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖))) = (𝑖 ∈ 𝐼 ↦ (((𝑏‘𝑖) ↑ 𝐿) · ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))
40	39	oveq2d 7436	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖)))) = ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ (((𝑏‘𝑖) ↑ 𝐿) · ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))
41		eqid 2728	. . . . . . . . . 10 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
42	14, 41	ringidval 20123	. . . . . . . . 9 ⊢ (1_r‘𝑆) = (0_g‘(mulGrp‘𝑆))
43	13	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → 𝑆 ∈ CRing)
44	43, 15	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (mulGrp‘𝑆) ∈ CMnd)
45	13	crngringd 20186	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Ring)
46	14	ringmgp 20179	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
47	45, 46	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
48	47	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → (mulGrp‘𝑆) ∈ Mnd)
49	32, 33, 48, 23, 29	mulgnn0cld 19050	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → ((𝑏‘𝑖) ↑ 𝐿) ∈ 𝐾)
50	32, 33, 48, 23, 31	mulgnn0cld 19050	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑖 ∈ 𝐼) → ((𝑏‘𝑖) ↑ (𝐴‘𝑖)) ∈ 𝐾)
51		eqidd 2729	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿)) = (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿)))
52		eqidd 2729	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))) = (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))
53	1	mptexd 7236	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿)) ∈ V)
54	53	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿)) ∈ V)
55		fvexd 6912	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (1_r‘𝑆) ∈ V)
56		funmpt 6591	. . . . . . . . . . 11 ⊢ Fun (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿))
57	56	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → Fun (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿)))
58	19	psrbagfsupp 21853	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} → 𝑏 finSupp 0)
59	18, 58	syl 17	. . . . . . . . . . 11 ⊢ (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} → 𝑏 finSupp 0)
60	59	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → 𝑏 finSupp 0)
61	22	feqmptd 6967	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → 𝑏 = (𝑖 ∈ 𝐼 ↦ (𝑏‘𝑖)))
62	61	oveq1d 7435	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (𝑏 supp 0) = ((𝑖 ∈ 𝐼 ↦ (𝑏‘𝑖)) supp 0))
63	62	eqimsscd 4037	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((𝑖 ∈ 𝐼 ↦ (𝑏‘𝑖)) supp 0) ⊆ (𝑏 supp 0))
64	32, 42, 33	mulg0 19030	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝐾 → (0 ↑ 𝑘) = (1_r‘𝑆))
65	64	adantl 481	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) ∧ 𝑘 ∈ 𝐾) → (0 ↑ 𝑘) = (1_r‘𝑆))
66		0zd 12601	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → 0 ∈ ℤ)
67	63, 65, 23, 29, 66	suppssov1 8203	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿)) supp (1_r‘𝑆)) ⊆ (𝑏 supp 0))
68	54, 55, 57, 60, 67	fsuppsssuppgd 9406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿)) finSupp (1_r‘𝑆))
69	1	mptexd 7236	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))) ∈ V)
70	69	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))) ∈ V)
71		funmpt 6591	. . . . . . . . . . 11 ⊢ Fun (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))
72	71	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → Fun (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))
73	63, 65, 23, 31, 66	suppssov1 8203	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))) supp (1_r‘𝑆)) ⊆ (𝑏 supp 0))
74	70, 55, 72, 60, 73	fsuppsssuppgd 9406	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))) finSupp (1_r‘𝑆))
75	32, 42, 35, 44, 2, 49, 50, 51, 52, 68, 74	gsummptfsadd 19879	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ (((𝑏‘𝑖) ↑ 𝐿) · ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) = (((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿))) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))
76		eqid 2728	. . . . . . . . . 10 ⊢ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} = {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}
77		mhphf.n	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
78	19, 76, 32, 33, 1, 47, 28, 77	mhphflem 41829	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿))) = (𝑁 ↑ 𝐿))
79	78	oveq1d 7435	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ 𝐿))) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) = ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))
80	40, 75, 79	3eqtrd 2772	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖)))) = ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))
81	80	oveq2d 7436	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖))))) = ((𝑋‘𝑏) · ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))
82		eqid 2728	. . . . . . . . . . 11 ⊢ (𝐼 mPoly 𝑈) = (𝐼 mPoly 𝑈)
83		eqid 2728	. . . . . . . . . . 11 ⊢ (Base‘𝑈) = (Base‘𝑈)
84		eqid 2728	. . . . . . . . . . 11 ⊢ (Base‘(𝐼 mPoly 𝑈)) = (Base‘(𝐼 mPoly 𝑈))
85		mhphf.h	. . . . . . . . . . . 12 ⊢ 𝐻 = (𝐼 mHomP 𝑈)
86		mhphf.u	. . . . . . . . . . . . . 14 ⊢ 𝑈 = (𝑆 ↾_s 𝑅)
87	86	ovexi 7454	. . . . . . . . . . . . 13 ⊢ 𝑈 ∈ V
88	87	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑈 ∈ V)
89		mhphf.x	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑋 ∈ (𝐻‘𝑁))
90	85, 82, 84, 1, 88, 77, 89	mhpmpl 22068	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐼 mPoly 𝑈)))
91	82, 83, 84, 19, 90	mplelf 21940	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:{ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑈))
92	86	subrgbas 20520	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 = (Base‘𝑈))
93	92, 26	eqsstrrd 4019	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (Base‘𝑈) ⊆ 𝐾)
94	24, 93	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑈) ⊆ 𝐾)
95	91, 94	fssd 6740	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:{ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}⟶𝐾)
96	95	ffvelcdmda 7094	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → (𝑋‘𝑏) ∈ 𝐾)
97	18, 96	sylan2 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (𝑋‘𝑏) ∈ 𝐾)
98	32, 33, 47, 77, 28	mulgnn0cld 19050	. . . . . . . 8 ⊢ (𝜑 → (𝑁 ↑ 𝐿) ∈ 𝐾)
99	98	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → (𝑁 ↑ 𝐿) ∈ 𝐾)
100	1	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝐼 ∈ 𝑉)
101	13	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑆 ∈ CRing)
102	5	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝐴 ∈ (𝐾 ↑_m 𝐼))
103		simpr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin})
104	19, 25, 14, 33, 100, 101, 102, 103	evlsvvvallem 41794	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))) ∈ 𝐾)
105	18, 104	sylan2 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))) ∈ 𝐾)
106	25, 34, 43, 97, 99, 105	crng12d 20198	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((𝑋‘𝑏) · ((𝑁 ↑ 𝐿) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) = ((𝑁 ↑ 𝐿) · ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))
107	81, 106	eqtrd 2768	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖))))) = ((𝑁 ↑ 𝐿) · ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))
108	107	mpteq2dva 5248	. . . 4 ⊢ (𝜑 → (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖)))))) = (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑁 ↑ 𝐿) · ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
109	108	oveq2d 7436	. . 3 ⊢ (𝜑 → (𝑆 Σ_g (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖))))))) = (𝑆 Σ_g (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑁 ↑ 𝐿) · ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))))
110		eqid 2728	. . . 4 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
111		ovex 7453	. . . . . . 7 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
112	111	rabex 5334	. . . . . 6 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∈ V
113	112	rabex 5334	. . . . 5 ⊢ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ∈ V
114	113	a1i 11	. . . 4 ⊢ (𝜑 → {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ∈ V)
115	45	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → 𝑆 ∈ Ring)
116	25, 34, 115, 96, 104	ringcld 20199	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}) → ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) ∈ 𝐾)
117	18, 116	sylan2 592	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁}) → ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))) ∈ 𝐾)
118		ssrab2 4075	. . . . . 6 ⊢ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ⊆ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
119		mptss 6046	. . . . . 6 ⊢ ({𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ⊆ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} → (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) ⊆ (𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))
120	118, 119	mp1i 13	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) ⊆ (𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))
121	19, 82, 86, 84, 25, 14, 33, 34, 1, 13, 24, 90, 5	evlsvvvallem2 41795	. . . . 5 ⊢ (𝜑 → (𝑏 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) finSupp (0_g‘𝑆))
122	120, 121	fsuppss 9407	. . . 4 ⊢ (𝜑 → (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))) finSupp (0_g‘𝑆))
123	25, 110, 34, 45, 114, 98, 117, 122	gsummulc2 20253	. . 3 ⊢ (𝜑 → (𝑆 Σ_g (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑁 ↑ 𝐿) · ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))) = ((𝑁 ↑ 𝐿) · (𝑆 Σ_g (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))))
124	109, 123	eqtrd 2768	. 2 ⊢ (𝜑 → (𝑆 Σ_g (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖))))))) = ((𝑁 ↑ 𝐿) · (𝑆 Σ_g (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))))
125		mhphf.q	. . 3 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
126	25	fvexi 6911	. . . . 5 ⊢ 𝐾 ∈ V
127	126	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
128	25, 34	ringcl 20190	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ 𝑗 ∈ 𝐾 ∧ 𝑘 ∈ 𝐾) → (𝑗 · 𝑘) ∈ 𝐾)
129	45, 128	syl3an1 1161	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐾 ∧ 𝑘 ∈ 𝐾) → (𝑗 · 𝑘) ∈ 𝐾)
130	129	3expb 1118	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐾 ∧ 𝑘 ∈ 𝐾)) → (𝑗 · 𝑘) ∈ 𝐾)
131		fconst6g 6786	. . . . . 6 ⊢ (𝐿 ∈ 𝐾 → (𝐼 × {𝐿}):𝐼⟶𝐾)
132	28, 131	syl 17	. . . . 5 ⊢ (𝜑 → (𝐼 × {𝐿}):𝐼⟶𝐾)
133		inidm 4219	. . . . 5 ⊢ (𝐼 ∩ 𝐼) = 𝐼
134	130, 132, 7, 1, 1, 133	off 7703	. . . 4 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘_f · 𝐴):𝐼⟶𝐾)
135	127, 1, 134	elmapdd 8860	. . 3 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘_f · 𝐴) ∈ (𝐾 ↑_m 𝐼))
136	125, 85, 86, 19, 76, 25, 14, 33, 34, 1, 13, 24, 77, 89, 135	evlsmhpvvval 41828	. 2 ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘_f · 𝐴)) = (𝑆 Σ_g (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (((𝐼 × {𝐿}) ∘_f · 𝐴)‘𝑖))))))))
137	125, 85, 86, 19, 76, 25, 14, 33, 34, 1, 13, 24, 77, 89, 5	evlsmhpvvval 41828	. . 3 ⊢ (𝜑 → ((𝑄‘𝑋)‘𝐴) = (𝑆 Σ_g (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖))))))))
138	137	oveq2d 7436	. 2 ⊢ (𝜑 → ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)) = ((𝑁 ↑ 𝐿) · (𝑆 Σ_g (𝑏 ∈ {𝑔 ∈ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∣ ((ℂ_fld ↾_s ℕ₀) Σ_g 𝑔) = 𝑁} ↦ ((𝑋‘𝑏) · ((mulGrp‘𝑆) Σ_g (𝑖 ∈ 𝐼 ↦ ((𝑏‘𝑖) ↑ (𝐴‘𝑖)))))))))
139	124, 136, 138	3eqtr4d 2778	1 ⊢ (𝜑 → ((𝑄‘𝑋)‘((𝐼 × {𝐿}) ∘_f · 𝐴)) = ((𝑁 ↑ 𝐿) · ((𝑄‘𝑋)‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 {crab 3429 Vcvv 3471 ⊆ wss 3947 {csn 4629 class class class wbr 5148 ↦ cmpt 5231 × cxp 5676 ^◡ccnv 5677 “ cima 5681 Fun wfun 6542 Fn wfn 6543 ⟶wf 6544 ‘cfv 6548 (class class class)co 7420 ∘_f cof 7683 supp csupp 8165 ↑_m cmap 8845 Fincfn 8964 finSupp cfsupp 9386 0cc0 11139 ℕcn 12243 ℕ₀cn0 12503 ℤcz 12589 Basecbs 17180 ↾_s cress 17209 ._rcmulr 17234 0_gc0g 17421 Σ_g cgsu 17422 Mndcmnd 18694 ._gcmg 19023 CMndccmn 19735 mulGrpcmgp 20074 1_rcur 20121 Ringcrg 20173 CRingccrg 20174 SubRingcsubrg 20506 ℂ_fldccnfld 21279 mPoly cmpl 21839 evalSub ces 22016 mHomP cmhp 22055

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-addf 11218

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7685 df-ofr 7686 df-om 7871 df-1st 7993 df-2nd 7994 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9387 df-sup 9466 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-fz 13518 df-fzo 13661 df-seq 14000 df-hash 14323 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-hom 17257 df-cco 17258 df-0g 17423 df-gsum 17424 df-prds 17429 df-pws 17431 df-mre 17566 df-mrc 17567 df-acs 17569 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-mhm 18740 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-mulg 19024 df-subg 19078 df-ghm 19168 df-cntz 19268 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-srg 20127 df-ring 20175 df-cring 20176 df-rhm 20411 df-subrng 20483 df-subrg 20508 df-lmod 20745 df-lss 20816 df-lsp 20856 df-cnfld 21280 df-assa 21787 df-asp 21788 df-ascl 21789 df-psr 21842 df-mvr 21843 df-mpl 21844 df-evls 22018 df-mhp 22062

This theorem is referenced by: mhphf2 41831 mhphf3 41832

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