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Theorem mhpmulcl 21089
Description: A product of homogeneous polynomials is a homogeneous polynomial whose degree is the sum of the degrees of the factors. Compare mdegmulle2 24977 (which shows less-than-or-equal instead of equal). (Contributed by SN, 22-Jul-2024.)
Hypotheses
Ref Expression
mhpmulcl.h 𝐻 = (𝐼 mHomP 𝑅)
mhpmulcl.y 𝑌 = (𝐼 mPoly 𝑅)
mhpmulcl.t · = (.r𝑌)
mhpmulcl.i (𝜑𝐼𝑉)
mhpmulcl.r (𝜑𝑅 ∈ Ring)
mhpmulcl.m (𝜑𝑀 ∈ ℕ0)
mhpmulcl.n (𝜑𝑁 ∈ ℕ0)
mhpmulcl.p (𝜑𝑃 ∈ (𝐻𝑀))
mhpmulcl.q (𝜑𝑄 ∈ (𝐻𝑁))
Assertion
Ref Expression
mhpmulcl (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)))

Proof of Theorem mhpmulcl
Dummy variables 𝑏 𝑑 𝑒 𝑖 𝑥 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mhpmulcl.y . . . . . . . 8 𝑌 = (𝐼 mPoly 𝑅)
2 eqid 2737 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
3 eqid 2737 . . . . . . . 8 (.r𝑅) = (.r𝑅)
4 mhpmulcl.t . . . . . . . 8 · = (.r𝑌)
5 eqid 2737 . . . . . . . 8 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} = { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}
6 mhpmulcl.h . . . . . . . . 9 𝐻 = (𝐼 mHomP 𝑅)
7 mhpmulcl.i . . . . . . . . 9 (𝜑𝐼𝑉)
8 mhpmulcl.r . . . . . . . . 9 (𝜑𝑅 ∈ Ring)
9 mhpmulcl.m . . . . . . . . 9 (𝜑𝑀 ∈ ℕ0)
10 mhpmulcl.p . . . . . . . . 9 (𝜑𝑃 ∈ (𝐻𝑀))
116, 1, 2, 7, 8, 9, 10mhpmpl 21084 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝑌))
12 mhpmulcl.n . . . . . . . . 9 (𝜑𝑁 ∈ ℕ0)
13 mhpmulcl.q . . . . . . . . 9 (𝜑𝑄 ∈ (𝐻𝑁))
146, 1, 2, 7, 8, 12, 13mhpmpl 21084 . . . . . . . 8 (𝜑𝑄 ∈ (Base‘𝑌))
151, 2, 3, 4, 5, 11, 14mplmul 20971 . . . . . . 7 (𝜑 → (𝑃 · 𝑄) = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑑} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑑f𝑒)))))))
1615adantr 484 . . . . . 6 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑃 · 𝑄) = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑑} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑑f𝑒)))))))
17 breq2 5057 . . . . . . . . . 10 (𝑑 = 𝑥 → (𝑐r𝑑𝑐r𝑥))
1817rabbidv 3390 . . . . . . . . 9 (𝑑 = 𝑥 → {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑑} = {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥})
19 fvoveq1 7236 . . . . . . . . . 10 (𝑑 = 𝑥 → (𝑄‘(𝑑f𝑒)) = (𝑄‘(𝑥f𝑒)))
2019oveq2d 7229 . . . . . . . . 9 (𝑑 = 𝑥 → ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑑f𝑒))) = ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))
2118, 20mpteq12dv 5140 . . . . . . . 8 (𝑑 = 𝑥 → (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑑} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑑f𝑒)))) = (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒)))))
2221oveq2d 7229 . . . . . . 7 (𝑑 = 𝑥 → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑑} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑑f𝑒))))) = (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))))
2322adantl 485 . . . . . 6 (((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ 𝑑 = 𝑥) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑑} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑑f𝑒))))) = (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))))
24 simpr 488 . . . . . 6 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
25 ovexd 7248 . . . . . 6 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))) ∈ V)
2616, 23, 24, 25fvmptd 6825 . . . . 5 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑃 · 𝑄)‘𝑥) = (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))))
2726neeq1d 3000 . . . 4 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((𝑃 · 𝑄)‘𝑥) ≠ (0g𝑅) ↔ (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))) ≠ (0g𝑅)))
28 simp-4l 783 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → 𝜑)
29 oveq2 7221 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑒 → ((ℂflds0) Σg 𝑐) = ((ℂflds0) Σg 𝑒))
3029eqeq1d 2739 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑒 → (((ℂflds0) Σg 𝑐) = 𝑀 ↔ ((ℂflds0) Σg 𝑒) = 𝑀))
3130necon3bbid 2978 . . . . . . . . . . . . . . 15 (𝑐 = 𝑒 → (¬ ((ℂflds0) Σg 𝑐) = 𝑀 ↔ ((ℂflds0) Σg 𝑒) ≠ 𝑀))
32 simplr 769 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥})
33 elrabi 3596 . . . . . . . . . . . . . . . 16 (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} → 𝑒 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
3432, 33syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → 𝑒 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
35 simpr 488 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → ((ℂflds0) Σg 𝑒) ≠ 𝑀)
3631, 34, 35elrabd 3604 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ¬ ((ℂflds0) Σg 𝑐) = 𝑀})
37 notrab 4226 . . . . . . . . . . . . . 14 ({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∖ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑀}) = {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ¬ ((ℂflds0) Σg 𝑐) = 𝑀}
3836, 37eleqtrrdi 2849 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → 𝑒 ∈ ({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∖ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑀}))
39 eqid 2737 . . . . . . . . . . . . . . 15 (Base‘𝑅) = (Base‘𝑅)
401, 39, 2, 5, 11mplelf 20960 . . . . . . . . . . . . . 14 (𝜑𝑃:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
41 eqid 2737 . . . . . . . . . . . . . . 15 (0g𝑅) = (0g𝑅)
426, 41, 5, 7, 8, 9, 10mhpdeg 21085 . . . . . . . . . . . . . 14 (𝜑 → (𝑃 supp (0g𝑅)) ⊆ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑀})
43 ovex 7246 . . . . . . . . . . . . . . . 16 (ℕ0m 𝐼) ∈ V
4443rabex 5225 . . . . . . . . . . . . . . 15 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V
4544a1i 11 . . . . . . . . . . . . . 14 (𝜑 → { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V)
46 fvexd 6732 . . . . . . . . . . . . . 14 (𝜑 → (0g𝑅) ∈ V)
4740, 42, 45, 46suppssr 7938 . . . . . . . . . . . . 13 ((𝜑𝑒 ∈ ({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∖ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑀})) → (𝑃𝑒) = (0g𝑅))
4828, 38, 47syl2anc 587 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → (𝑃𝑒) = (0g𝑅))
4948oveq1d 7228 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))) = ((0g𝑅)(.r𝑅)(𝑄‘(𝑥f𝑒))))
508ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → 𝑅 ∈ Ring)
5114ad4antr 732 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → 𝑄 ∈ (Base‘𝑌))
521, 39, 2, 5, 51mplelf 20960 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → 𝑄:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
53 simp-4r 784 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → 𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
54 eqid 2737 . . . . . . . . . . . . . . . 16 {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} = {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}
555, 54psrbagconcl 20893 . . . . . . . . . . . . . . 15 ((𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑥f𝑒) ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥})
5653, 32, 55syl2anc 587 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → (𝑥f𝑒) ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥})
57 elrabi 3596 . . . . . . . . . . . . . 14 ((𝑥f𝑒) ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} → (𝑥f𝑒) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
5856, 57syl 17 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → (𝑥f𝑒) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
5952, 58ffvelrnd 6905 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → (𝑄‘(𝑥f𝑒)) ∈ (Base‘𝑅))
6039, 3, 41ringlz 19605 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑄‘(𝑥f𝑒)) ∈ (Base‘𝑅)) → ((0g𝑅)(.r𝑅)(𝑄‘(𝑥f𝑒))) = (0g𝑅))
6150, 59, 60syl2anc 587 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → ((0g𝑅)(.r𝑅)(𝑄‘(𝑥f𝑒))) = (0g𝑅))
6249, 61eqtrd 2777 . . . . . . . . . 10 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg 𝑒) ≠ 𝑀) → ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))) = (0g𝑅))
63 simp-4l 783 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → 𝜑)
64 oveq2 7221 . . . . . . . . . . . . . . . . 17 (𝑐 = (𝑥f𝑒) → ((ℂflds0) Σg 𝑐) = ((ℂflds0) Σg (𝑥f𝑒)))
6564eqeq1d 2739 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑥f𝑒) → (((ℂflds0) Σg 𝑐) = 𝑁 ↔ ((ℂflds0) Σg (𝑥f𝑒)) = 𝑁))
6665necon3bbid 2978 . . . . . . . . . . . . . . 15 (𝑐 = (𝑥f𝑒) → (¬ ((ℂflds0) Σg 𝑐) = 𝑁 ↔ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁))
67 simp-4r 784 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → 𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
68 simplr 769 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥})
6967, 68, 55syl2anc 587 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → (𝑥f𝑒) ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥})
7069, 57syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → (𝑥f𝑒) ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
71 simpr 488 . . . . . . . . . . . . . . 15 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁)
7266, 70, 71elrabd 3604 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → (𝑥f𝑒) ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ¬ ((ℂflds0) Σg 𝑐) = 𝑁})
73 notrab 4226 . . . . . . . . . . . . . 14 ({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∖ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑁}) = {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ¬ ((ℂflds0) Σg 𝑐) = 𝑁}
7472, 73eleqtrrdi 2849 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → (𝑥f𝑒) ∈ ({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∖ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑁}))
751, 39, 2, 5, 14mplelf 20960 . . . . . . . . . . . . . 14 (𝜑𝑄:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
766, 41, 5, 7, 8, 12, 13mhpdeg 21085 . . . . . . . . . . . . . 14 (𝜑 → (𝑄 supp (0g𝑅)) ⊆ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑁})
7775, 76, 45, 46suppssr 7938 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥f𝑒) ∈ ({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∖ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑁})) → (𝑄‘(𝑥f𝑒)) = (0g𝑅))
7863, 74, 77syl2anc 587 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → (𝑄‘(𝑥f𝑒)) = (0g𝑅))
7978oveq2d 7229 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))) = ((𝑃𝑒)(.r𝑅)(0g𝑅)))
808ad4antr 732 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → 𝑅 ∈ Ring)
8111ad4antr 732 . . . . . . . . . . . . . 14 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → 𝑃 ∈ (Base‘𝑌))
821, 39, 2, 5, 81mplelf 20960 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → 𝑃:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
8333adantl 485 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑒 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
8483adantr 484 . . . . . . . . . . . . 13 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → 𝑒 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
8582, 84ffvelrnd 6905 . . . . . . . . . . . 12 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → (𝑃𝑒) ∈ (Base‘𝑅))
8639, 3, 41ringrz 19606 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑃𝑒) ∈ (Base‘𝑅)) → ((𝑃𝑒)(.r𝑅)(0g𝑅)) = (0g𝑅))
8780, 85, 86syl2anc 587 . . . . . . . . . . 11 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → ((𝑃𝑒)(.r𝑅)(0g𝑅)) = (0g𝑅))
8879, 87eqtrd 2777 . . . . . . . . . 10 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) → ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))) = (0g𝑅))
89 nn0subm 20418 . . . . . . . . . . . . . . . 16 0 ∈ (SubMnd‘ℂfld)
90 eqid 2737 . . . . . . . . . . . . . . . . 17 (ℂflds0) = (ℂflds0)
9190submbas 18241 . . . . . . . . . . . . . . . 16 (ℕ0 ∈ (SubMnd‘ℂfld) → ℕ0 = (Base‘(ℂflds0)))
9289, 91ax-mp 5 . . . . . . . . . . . . . . 15 0 = (Base‘(ℂflds0))
93 cnfld0 20387 . . . . . . . . . . . . . . . . 17 0 = (0g‘ℂfld)
9490, 93subm0 18242 . . . . . . . . . . . . . . . 16 (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂflds0)))
9589, 94ax-mp 5 . . . . . . . . . . . . . . 15 0 = (0g‘(ℂflds0))
96 nn0ex 12096 . . . . . . . . . . . . . . . 16 0 ∈ V
97 cnfldadd 20368 . . . . . . . . . . . . . . . . 17 + = (+g‘ℂfld)
9890, 97ressplusg 16834 . . . . . . . . . . . . . . . 16 (ℕ0 ∈ V → + = (+g‘(ℂflds0)))
9996, 98ax-mp 5 . . . . . . . . . . . . . . 15 + = (+g‘(ℂflds0))
100 cnring 20385 . . . . . . . . . . . . . . . . . 18 fld ∈ Ring
101 ringcmn 19599 . . . . . . . . . . . . . . . . . 18 (ℂfld ∈ Ring → ℂfld ∈ CMnd)
102100, 101ax-mp 5 . . . . . . . . . . . . . . . . 17 fld ∈ CMnd
10390submcmn 19223 . . . . . . . . . . . . . . . . 17 ((ℂfld ∈ CMnd ∧ ℕ0 ∈ (SubMnd‘ℂfld)) → (ℂflds0) ∈ CMnd)
104102, 89, 103mp2an 692 . . . . . . . . . . . . . . . 16 (ℂflds0) ∈ CMnd
105104a1i 11 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (ℂflds0) ∈ CMnd)
1067ad3antrrr 730 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝐼𝑉)
1075psrbagf 20877 . . . . . . . . . . . . . . . 16 (𝑒 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑒:𝐼⟶ℕ0)
10883, 107syl 17 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑒:𝐼⟶ℕ0)
109 simpllr 776 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
1105psrbagf 20877 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑥:𝐼⟶ℕ0)
111109, 110syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑥:𝐼⟶ℕ0)
112111ffnd 6546 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑥 Fn 𝐼)
113108ffnd 6546 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑒 Fn 𝐼)
114 inidm 4133 . . . . . . . . . . . . . . . . 17 (𝐼𝐼) = 𝐼
115 eqidd 2738 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → (𝑥𝑖) = (𝑥𝑖))
116 eqidd 2738 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → (𝑒𝑖) = (𝑒𝑖))
117112, 113, 106, 106, 114, 115, 116offval 7477 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑥f𝑒) = (𝑖𝐼 ↦ ((𝑥𝑖) − (𝑒𝑖))))
118 simpl 486 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → (((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}))
119 simplr 769 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥})
120 breq1 5056 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑒 → (𝑐r𝑥𝑒r𝑥))
121120elrab 3602 . . . . . . . . . . . . . . . . . . . 20 (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↔ (𝑒 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∧ 𝑒r𝑥))
122121simprbi 500 . . . . . . . . . . . . . . . . . . 19 (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} → 𝑒r𝑥)
123119, 122syl 17 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → 𝑒r𝑥)
124 simpr 488 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → 𝑖𝐼)
125113, 112, 106, 106, 114, 116, 115ofrval 7480 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑒r𝑥𝑖𝐼) → (𝑒𝑖) ≤ (𝑥𝑖))
126118, 123, 124, 125syl3anc 1373 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → (𝑒𝑖) ≤ (𝑥𝑖))
127108ffvelrnda 6904 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → (𝑒𝑖) ∈ ℕ0)
128111ffvelrnda 6904 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → (𝑥𝑖) ∈ ℕ0)
129 nn0sub 12140 . . . . . . . . . . . . . . . . . 18 (((𝑒𝑖) ∈ ℕ0 ∧ (𝑥𝑖) ∈ ℕ0) → ((𝑒𝑖) ≤ (𝑥𝑖) ↔ ((𝑥𝑖) − (𝑒𝑖)) ∈ ℕ0))
130127, 128, 129syl2anc 587 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → ((𝑒𝑖) ≤ (𝑥𝑖) ↔ ((𝑥𝑖) − (𝑒𝑖)) ∈ ℕ0))
131126, 130mpbid 235 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → ((𝑥𝑖) − (𝑒𝑖)) ∈ ℕ0)
132117, 131fmpt3d 6933 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑥f𝑒):𝐼⟶ℕ0)
133108ffund 6549 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → Fun 𝑒)
134 c0ex 10827 . . . . . . . . . . . . . . . . . . . 20 0 ∈ V
135106, 134jctir 524 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝐼𝑉 ∧ 0 ∈ V))
136 frnsuppeq 7917 . . . . . . . . . . . . . . . . . . 19 ((𝐼𝑉 ∧ 0 ∈ V) → (𝑒:𝐼⟶ℕ0 → (𝑒 supp 0) = (𝑒 “ (ℕ0 ∖ {0}))))
137135, 108, 136sylc 65 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒 supp 0) = (𝑒 “ (ℕ0 ∖ {0})))
138 dfn2 12103 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℕ0 ∖ {0})
139138imaeq2i 5927 . . . . . . . . . . . . . . . . . 18 (𝑒 “ ℕ) = (𝑒 “ (ℕ0 ∖ {0}))
140137, 139eqtr4di 2796 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒 supp 0) = (𝑒 “ ℕ))
1415psrbag 20876 . . . . . . . . . . . . . . . . . . . 20 (𝐼𝑉 → (𝑒 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↔ (𝑒:𝐼⟶ℕ0 ∧ (𝑒 “ ℕ) ∈ Fin)))
142106, 141syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↔ (𝑒:𝐼⟶ℕ0 ∧ (𝑒 “ ℕ) ∈ Fin)))
14383, 142mpbid 235 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒:𝐼⟶ℕ0 ∧ (𝑒 “ ℕ) ∈ Fin))
144143simprd 499 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒 “ ℕ) ∈ Fin)
145140, 144eqeltrd 2838 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒 supp 0) ∈ Fin)
14683elexd 3428 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑒 ∈ V)
147 isfsupp 8989 . . . . . . . . . . . . . . . . 17 ((𝑒 ∈ V ∧ 0 ∈ V) → (𝑒 finSupp 0 ↔ (Fun 𝑒 ∧ (𝑒 supp 0) ∈ Fin)))
148146, 134, 147sylancl 589 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒 finSupp 0 ↔ (Fun 𝑒 ∧ (𝑒 supp 0) ∈ Fin)))
149133, 145, 148mpbir2and 713 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑒 finSupp 0)
150112, 113, 106, 106offun 7482 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → Fun (𝑥f𝑒))
1515psrbagfsupp 20879 . . . . . . . . . . . . . . . . . . 19 (𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} → 𝑥 finSupp 0)
152109, 151syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑥 finSupp 0)
153152, 149fsuppunfi 9005 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((𝑥 supp 0) ∪ (𝑒 supp 0)) ∈ Fin)
154 0nn0 12105 . . . . . . . . . . . . . . . . . . 19 0 ∈ ℕ0
155154a1i 11 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 0 ∈ ℕ0)
156 0m0e0 11950 . . . . . . . . . . . . . . . . . . 19 (0 − 0) = 0
157156a1i 11 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (0 − 0) = 0)
158106, 155, 111, 108, 157suppofssd 7945 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((𝑥f𝑒) supp 0) ⊆ ((𝑥 supp 0) ∪ (𝑒 supp 0)))
159153, 158ssfid 8898 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((𝑥f𝑒) supp 0) ∈ Fin)
160 ovexd 7248 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑥f𝑒) ∈ V)
161 isfsupp 8989 . . . . . . . . . . . . . . . . 17 (((𝑥f𝑒) ∈ V ∧ 0 ∈ V) → ((𝑥f𝑒) finSupp 0 ↔ (Fun (𝑥f𝑒) ∧ ((𝑥f𝑒) supp 0) ∈ Fin)))
162160, 134, 161sylancl 589 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((𝑥f𝑒) finSupp 0 ↔ (Fun (𝑥f𝑒) ∧ ((𝑥f𝑒) supp 0) ∈ Fin)))
163150, 159, 162mpbir2and 713 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑥f𝑒) finSupp 0)
16492, 95, 99, 105, 106, 108, 132, 149, 163gsumadd 19308 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((ℂflds0) Σg (𝑒f + (𝑥f𝑒))) = (((ℂflds0) Σg 𝑒) + ((ℂflds0) Σg (𝑥f𝑒))))
165108ffvelrnda 6904 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → (𝑒𝑏) ∈ ℕ0)
166165nn0cnd 12152 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → (𝑒𝑏) ∈ ℂ)
167111ffvelrnda 6904 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → (𝑥𝑏) ∈ ℕ0)
168167nn0cnd 12152 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → (𝑥𝑏) ∈ ℂ)
169166, 168pncan3d 11192 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → ((𝑒𝑏) + ((𝑥𝑏) − (𝑒𝑏))) = (𝑥𝑏))
170169mpteq2dva 5150 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑏𝐼 ↦ ((𝑒𝑏) + ((𝑥𝑏) − (𝑒𝑏)))) = (𝑏𝐼 ↦ (𝑥𝑏)))
171 fvexd 6732 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → (𝑒𝑏) ∈ V)
172 ovexd 7248 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → ((𝑥𝑏) − (𝑒𝑏)) ∈ V)
173108feqmptd 6780 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑒 = (𝑏𝐼 ↦ (𝑒𝑏)))
174111feqmptd 6780 . . . . . . . . . . . . . . . . . 18 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑥 = (𝑏𝐼 ↦ (𝑥𝑏)))
175106, 167, 165, 174, 173offval2 7488 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑥f𝑒) = (𝑏𝐼 ↦ ((𝑥𝑏) − (𝑒𝑏))))
176106, 171, 172, 173, 175offval2 7488 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒f + (𝑥f𝑒)) = (𝑏𝐼 ↦ ((𝑒𝑏) + ((𝑥𝑏) − (𝑒𝑏)))))
177170, 176, 1743eqtr4d 2787 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒f + (𝑥f𝑒)) = 𝑥)
178177oveq2d 7229 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((ℂflds0) Σg (𝑒f + (𝑥f𝑒))) = ((ℂflds0) Σg 𝑥))
179164, 178eqtr3d 2779 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (((ℂflds0) Σg 𝑒) + ((ℂflds0) Σg (𝑥f𝑒))) = ((ℂflds0) Σg 𝑥))
180 simplr 769 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁))
181179, 180eqnetrd 3008 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (((ℂflds0) Σg 𝑒) + ((ℂflds0) Σg (𝑥f𝑒))) ≠ (𝑀 + 𝑁))
182 oveq12 7222 . . . . . . . . . . . . . 14 ((((ℂflds0) Σg 𝑒) = 𝑀 ∧ ((ℂflds0) Σg (𝑥f𝑒)) = 𝑁) → (((ℂflds0) Σg 𝑒) + ((ℂflds0) Σg (𝑥f𝑒))) = (𝑀 + 𝑁))
183182a1i 11 . . . . . . . . . . . . 13 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((((ℂflds0) Σg 𝑒) = 𝑀 ∧ ((ℂflds0) Σg (𝑥f𝑒)) = 𝑁) → (((ℂflds0) Σg 𝑒) + ((ℂflds0) Σg (𝑥f𝑒))) = (𝑀 + 𝑁)))
184183necon3ad 2953 . . . . . . . . . . . 12 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((((ℂflds0) Σg 𝑒) + ((ℂflds0) Σg (𝑥f𝑒))) ≠ (𝑀 + 𝑁) → ¬ (((ℂflds0) Σg 𝑒) = 𝑀 ∧ ((ℂflds0) Σg (𝑥f𝑒)) = 𝑁)))
185181, 184mpd 15 . . . . . . . . . . 11 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ¬ (((ℂflds0) Σg 𝑒) = 𝑀 ∧ ((ℂflds0) Σg (𝑥f𝑒)) = 𝑁))
186 neorian 3036 . . . . . . . . . . 11 ((((ℂflds0) Σg 𝑒) ≠ 𝑀 ∨ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) ↔ ¬ (((ℂflds0) Σg 𝑒) = 𝑀 ∧ ((ℂflds0) Σg (𝑥f𝑒)) = 𝑁))
187185, 186sylibr 237 . . . . . . . . . 10 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (((ℂflds0) Σg 𝑒) ≠ 𝑀 ∨ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁))
18862, 88, 187mpjaodan 959 . . . . . . . . 9 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))) = (0g𝑅))
189188mpteq2dva 5150 . . . . . . . 8 (((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒)))) = (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ (0g𝑅)))
190189oveq2d 7229 . . . . . . 7 (((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))) = (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ (0g𝑅))))
191 ringmnd 19572 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
1928, 191syl 17 . . . . . . . . 9 (𝜑𝑅 ∈ Mnd)
193192ad2antrr 726 . . . . . . . 8 (((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → 𝑅 ∈ Mnd)
19444rabex 5225 . . . . . . . 8 {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ∈ V
19541gsumz 18262 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ∈ V) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ (0g𝑅))) = (0g𝑅))
196193, 194, 195sylancl 589 . . . . . . 7 (((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ (0g𝑅))) = (0g𝑅))
197190, 196eqtrd 2777 . . . . . 6 (((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))) = (0g𝑅))
198197ex 416 . . . . 5 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))) = (0g𝑅)))
199198necon1d 2962 . . . 4 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))) ≠ (0g𝑅) → ((ℂflds0) Σg 𝑥) = (𝑀 + 𝑁)))
20027, 199sylbid 243 . . 3 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((𝑃 · 𝑄)‘𝑥) ≠ (0g𝑅) → ((ℂflds0) Σg 𝑥) = (𝑀 + 𝑁)))
201200ralrimiva 3105 . 2 (𝜑 → ∀𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} (((𝑃 · 𝑄)‘𝑥) ≠ (0g𝑅) → ((ℂflds0) Σg 𝑥) = (𝑀 + 𝑁)))
2029, 12nn0addcld 12154 . . 3 (𝜑 → (𝑀 + 𝑁) ∈ ℕ0)
2031mplring 20980 . . . . 5 ((𝐼𝑉𝑅 ∈ Ring) → 𝑌 ∈ Ring)
2047, 8, 203syl2anc 587 . . . 4 (𝜑𝑌 ∈ Ring)
2052, 4ringcl 19579 . . . 4 ((𝑌 ∈ Ring ∧ 𝑃 ∈ (Base‘𝑌) ∧ 𝑄 ∈ (Base‘𝑌)) → (𝑃 · 𝑄) ∈ (Base‘𝑌))
206204, 11, 14, 205syl3anc 1373 . . 3 (𝜑 → (𝑃 · 𝑄) ∈ (Base‘𝑌))
2076, 1, 2, 41, 5, 7, 8, 202, 206ismhp3 21083 . 2 (𝜑 → ((𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)) ↔ ∀𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} (((𝑃 · 𝑄)‘𝑥) ≠ (0g𝑅) → ((ℂflds0) Σg 𝑥) = (𝑀 + 𝑁))))
208201, 207mpbird 260 1 (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 847   = wceq 1543  wcel 2110  wne 2940  wral 3061  {crab 3065  Vcvv 3408  cdif 3863  cun 3864  {csn 4541   class class class wbr 5053  cmpt 5135  ccnv 5550  cima 5554  Fun wfun 6374  wf 6376  cfv 6380  (class class class)co 7213  f cof 7467  r cofr 7468   supp csupp 7903  m cmap 8508  Fincfn 8626   finSupp cfsupp 8985  0cc0 10729   + caddc 10732  cle 10868  cmin 11062  cn 11830  0cn0 12090  Basecbs 16760  s cress 16784  +gcplusg 16802  .rcmulr 16803  0gc0g 16944   Σg cgsu 16945  Mndcmnd 18173  SubMndcsubmnd 18217  CMndccmn 19170  Ringcrg 19562  fldccnfld 20363   mPoly cmpl 20865   mHomP cmhp 21069
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-addf 10808  ax-mulf 10809
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-of 7469  df-ofr 7470  df-om 7645  df-1st 7761  df-2nd 7762  df-supp 7904  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-er 8391  df-map 8510  df-pm 8511  df-ixp 8579  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-fsupp 8986  df-oi 9126  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-z 12177  df-dec 12294  df-uz 12439  df-fz 13096  df-fzo 13239  df-seq 13575  df-hash 13897  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-mulr 16816  df-starv 16817  df-sca 16818  df-vsca 16819  df-tset 16821  df-ple 16822  df-ds 16824  df-unif 16825  df-0g 16946  df-gsum 16947  df-mre 17089  df-mrc 17090  df-acs 17092  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-mhm 18218  df-submnd 18219  df-grp 18368  df-minusg 18369  df-mulg 18489  df-subg 18540  df-ghm 18620  df-cntz 18711  df-cmn 19172  df-abl 19173  df-mgp 19505  df-ur 19517  df-ring 19564  df-cring 19565  df-subrg 19798  df-cnfld 20364  df-psr 20868  df-mpl 20870  df-mhp 21073
This theorem is referenced by:  mhppwdeg  21090
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