mhphf - Mathbox for Steven Nguyen

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Theorem mhphf 41700

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 8840	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ (𝐾 ↑_m 𝐼) → 𝐴:𝐼⟶𝐾)
7	5, 6	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴:𝐼⟶𝐾)
8	7	ffnd 6709	. . . . . . . . . . . . . 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 2725	. . . . . . . . . . . . 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 7690	. . . . . . . . . . . 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 7418	. . . . . . . . . . 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 2724	. . . . . . . . . . . . . . 15 ⊢ (mulGrp‘𝑆) = (mulGrp‘𝑆)
15	14	crngmgp 20142	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ CRing → (mulGrp‘𝑆) ∈ CMnd)
16	13, 15	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ CMnd)
17	16	ad2antrr 723	. . . . . . . . . . . 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 3670	. . . . . . . . . . . . . . 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 2724	. . . . . . . . . . . . . . . 16 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} = {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}
20	19	psrbagf 21801	. . . . . . . . . . . . . . 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 7077	. . . . . . . . . . . 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 20470	. . . . . . . . . . . . . . 15 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐾)
27	24, 26	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑅 ⊆ 𝐾)
28	27, 3	sseldd 3976	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ 𝐾)
29	28	ad2antrr 723	. . . . . . . . . . . 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 7077	. . . . . . . . . . . 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 20041	. . . . . . . . . . . . 13 ⊢ 𝐾 = (Base‘(mulGrp‘𝑆))
33		mhphf.e	. . . . . . . . . . . . 13 ⊢ ↑ = (._g‘(mulGrp‘𝑆))
34		mhphf.m	. . . . . . . . . . . . . 14 ⊢ · = (._r‘𝑆)
35	14, 34	mgpplusg 20039	. . . . . . . . . . . . 13 ⊢ · = (+_g‘(mulGrp‘𝑆))
36	32, 33, 35	mulgnn0di 19741	. . . . . . . . . . . 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 1369	. . . . . . . . . . 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 2764	. . . . . . . . . 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 5239	. . . . . . . . 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 7418	. . . . . . . 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 2724	. . . . . . . . . 10 ⊢ (1_r‘𝑆) = (1_r‘𝑆)
42	14, 41	ringidval 20084	. . . . . . . . 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 20147	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ Ring)
46	14	ringmgp 20140	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ Ring → (mulGrp‘𝑆) ∈ Mnd)
47	45, 46	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (mulGrp‘𝑆) ∈ Mnd)
48	47	ad2antrr 723	. . . . . . . . . 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 19018	. . . . . . . . 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 19018	. . . . . . . . 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 2725	. . . . . . . . 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 2725	. . . . . . . . 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 7218	. . . . . . . . . . 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 6897	. . . . . . . . . 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 6577	. . . . . . . . . . 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 21803	. . . . . . . . . . . 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 6951	. . . . . . . . . . . . 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 7417	. . . . . . . . . . . 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 4032	. . . . . . . . . . 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 18998	. . . . . . . . . . . 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 12569	. . . . . . . . . . 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 8178	. . . . . . . . . 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 41599	. . . . . . . . 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 7218	. . . . . . . . . . 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 6577	. . . . . . . . . . 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 8178	. . . . . . . . . 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 41599	. . . . . . . . 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 19840	. . . . . . . 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 2724	. . . . . . . . . 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 41699	. . . . . . . . 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 7417	. . . . . . . 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 2768	. . . . . . 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 7418	. . . . . 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 2724	. . . . . . . . . . 11 ⊢ (𝐼 mPoly 𝑈) = (𝐼 mPoly 𝑈)
83		eqid 2724	. . . . . . . . . . 11 ⊢ (Base‘𝑈) = (Base‘𝑈)
84		eqid 2724	. . . . . . . . . . 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 7436	. . . . . . . . . . . . 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 22016	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑋 ∈ (Base‘(𝐼 mPoly 𝑈)))
91	82, 83, 84, 19, 90	mplelf 21888	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋:{ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}⟶(Base‘𝑈))
92	86	subrgbas 20479	. . . . . . . . . . . 12 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 = (Base‘𝑈))
93	92, 26	eqsstrrd 4014	. . . . . . . . . . 11 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (Base‘𝑈) ⊆ 𝐾)
94	24, 93	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (Base‘𝑈) ⊆ 𝐾)
95	91, 94	fssd 6726	. . . . . . . . 9 ⊢ (𝜑 → 𝑋:{ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin}⟶𝐾)
96	95	ffvelcdmda 7077	. . . . . . . 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 19018	. . . . . . . 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 41664	. . . . . . . 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 41621	. . . . . 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 2764	. . . . 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 5239	. . . 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 7418	. . 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 2724	. . . 4 ⊢ (0_g‘𝑆) = (0_g‘𝑆)
111		ovex 7435	. . . . . . 7 ⊢ (ℕ₀ ↑_m 𝐼) ∈ V
112	111	rabex 5323	. . . . . 6 ⊢ {ℎ ∈ (ℕ₀ ↑_m 𝐼) ∣ (^◡ℎ “ ℕ) ∈ Fin} ∈ V
113	112	rabex 5323	. . . . 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 20158	. . . . 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 4070	. . . . . 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 6033	. . . . . 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 41665	. . . . 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 41600	. . . 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 20212	. . 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 2764	. 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 6896	. . . . 5 ⊢ 𝐾 ∈ V
127	126	a1i 11	. . . 4 ⊢ (𝜑 → 𝐾 ∈ V)
128	25, 34	ringcl 20151	. . . . . . 7 ⊢ ((𝑆 ∈ Ring ∧ 𝑗 ∈ 𝐾 ∧ 𝑘 ∈ 𝐾) → (𝑗 · 𝑘) ∈ 𝐾)
129	45, 128	syl3an1 1160	. . . . . 6 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝐾 ∧ 𝑘 ∈ 𝐾) → (𝑗 · 𝑘) ∈ 𝐾)
130	129	3expb 1117	. . . . 5 ⊢ ((𝜑 ∧ (𝑗 ∈ 𝐾 ∧ 𝑘 ∈ 𝐾)) → (𝑗 · 𝑘) ∈ 𝐾)
131		fconst6g 6771	. . . . . 6 ⊢ (𝐿 ∈ 𝐾 → (𝐼 × {𝐿}):𝐼⟶𝐾)
132	28, 131	syl 17	. . . . 5 ⊢ (𝜑 → (𝐼 × {𝐿}):𝐼⟶𝐾)
133		inidm 4211	. . . . 5 ⊢ (𝐼 ∩ 𝐼) = 𝐼
134	130, 132, 7, 1, 1, 133	off 7682	. . . 4 ⊢ (𝜑 → ((𝐼 × {𝐿}) ∘_f · 𝐴):𝐼⟶𝐾)
135	127, 1, 134	elmapdd 8832	. . 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 41698	. 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 41698	. . 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 7418	. 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 2774	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 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 ⊆ wss 3941 {csn 4621 class class class wbr 5139 ↦ cmpt 5222 × cxp 5665 ^◡ccnv 5666 “ cima 5670 Fun wfun 6528 Fn wfn 6529 ⟶wf 6530 ‘cfv 6534 (class class class)co 7402 ∘_f cof 7662 supp csupp 8141 ↑_m cmap 8817 Fincfn 8936 finSupp cfsupp 9358 0cc0 11107 ℕcn 12211 ℕ₀cn0 12471 ℤcz 12557 Basecbs 17149 ↾_s cress 17178 ._rcmulr 17203 0_gc0g 17390 Σ_g cgsu 17391 Mndcmnd 18663 ._gcmg 18991 CMndccmn 19696 mulGrpcmgp 20035 1_rcur 20082 Ringcrg 20134 CRingccrg 20135 SubRingcsubrg 20465 ℂ_fldccnfld 21234 mPoly cmpl 21789 evalSub ces 21964 mHomP cmhp 22003

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5276 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-addf 11186 ax-mulf 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-tp 4626 df-op 4628 df-uni 4901 df-int 4942 df-iun 4990 df-iin 4991 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-se 5623 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-isom 6543 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-of 7664 df-ofr 7665 df-om 7850 df-1st 7969 df-2nd 7970 df-supp 8142 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8700 df-map 8819 df-pm 8820 df-ixp 8889 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-fsupp 9359 df-sup 9434 df-oi 9502 df-card 9931 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-nn 12212 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12472 df-z 12558 df-dec 12677 df-uz 12822 df-fz 13486 df-fzo 13629 df-seq 13968 df-hash 14292 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-0g 17392 df-gsum 17393 df-prds 17398 df-pws 17400 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-mhm 18709 df-submnd 18710 df-grp 18862 df-minusg 18863 df-sbg 18864 df-mulg 18992 df-subg 19046 df-ghm 19135 df-cntz 19229 df-cmn 19698 df-abl 19699 df-mgp 20036 df-rng 20054 df-ur 20083 df-srg 20088 df-ring 20136 df-cring 20137 df-rhm 20370 df-subrng 20442 df-subrg 20467 df-lmod 20704 df-lss 20775 df-lsp 20815 df-cnfld 21235 df-assa 21737 df-asp 21738 df-ascl 21739 df-psr 21792 df-mvr 21793 df-mpl 21794 df-evls 21966 df-mhp 22010

This theorem is referenced by: mhphf2 41701 mhphf3 41702

W3C validator