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Theorem mhpmulcl 21539
Description: A product of homogeneous polynomials is a homogeneous polynomial whose degree is the sum of the degrees of the factors. Compare mdegmulle2 25444 (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 2736 . . . . . . . 8 (Base‘𝑌) = (Base‘𝑌)
3 eqid 2736 . . . . . . . 8 (.r𝑅) = (.r𝑅)
4 mhpmulcl.t . . . . . . . 8 · = (.r𝑌)
5 eqid 2736 . . . . . . . 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 21534 . . . . . . . 8 (𝜑𝑃 ∈ (Base‘𝑌))
12 mhpmulcl.n . . . . . . . . 9 (𝜑𝑁 ∈ ℕ0)
13 mhpmulcl.q . . . . . . . . 9 (𝜑𝑄 ∈ (𝐻𝑁))
146, 1, 2, 7, 8, 12, 13mhpmpl 21534 . . . . . . . 8 (𝜑𝑄 ∈ (Base‘𝑌))
151, 2, 3, 4, 5, 11, 14mplmul 21415 . . . . . . 7 (𝜑 → (𝑃 · 𝑄) = (𝑑 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ↦ (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑑} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑑f𝑒)))))))
1615adantr 481 . . . . . 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 5109 . . . . . . . . . 10 (𝑑 = 𝑥 → (𝑐r𝑑𝑐r𝑥))
1817rabbidv 3415 . . . . . . . . 9 (𝑑 = 𝑥 → {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑑} = {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥})
19 fvoveq1 7380 . . . . . . . . . 10 (𝑑 = 𝑥 → (𝑄‘(𝑑f𝑒)) = (𝑄‘(𝑥f𝑒)))
2019oveq2d 7373 . . . . . . . . 9 (𝑑 = 𝑥 → ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑑f𝑒))) = ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))
2118, 20mpteq12dv 5196 . . . . . . . 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 7373 . . . . . . 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 482 . . . . . 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 485 . . . . . 6 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → 𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
25 ovexd 7392 . . . . . 6 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))) ∈ V)
2616, 23, 24, 25fvmptd 6955 . . . . 5 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → ((𝑃 · 𝑄)‘𝑥) = (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))))))
2726neeq1d 3003 . . . 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 781 . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑒 → ((ℂflds0) Σg 𝑐) = ((ℂflds0) Σg 𝑒))
3029eqeq1d 2738 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑒 → (((ℂflds0) Σg 𝑐) = 𝑀 ↔ ((ℂflds0) Σg 𝑒) = 𝑀))
3130necon3bbid 2981 . . . . . . . . . . . . . . 15 (𝑐 = 𝑒 → (¬ ((ℂflds0) Σg 𝑐) = 𝑀 ↔ ((ℂflds0) Σg 𝑒) ≠ 𝑀))
32 simplr 767 . . . . . . . . . . . . . . . 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 3639 . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . 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 3647 . . . . . . . . . . . . . 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 4271 . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . 15 (Base‘𝑅) = (Base‘𝑅)
401, 39, 2, 5, 11mplelf 21404 . . . . . . . . . . . . . 14 (𝜑𝑃:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
41 eqid 2736 . . . . . . . . . . . . . . 15 (0g𝑅) = (0g𝑅)
426, 41, 5, 7, 8, 9, 10mhpdeg 21535 . . . . . . . . . . . . . 14 (𝜑 → (𝑃 supp (0g𝑅)) ⊆ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑀})
43 ovex 7390 . . . . . . . . . . . . . . . 16 (ℕ0m 𝐼) ∈ V
4443rabex 5289 . . . . . . . . . . . . . . 15 { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V
4544a1i 11 . . . . . . . . . . . . . 14 (𝜑 → { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∈ V)
46 fvexd 6857 . . . . . . . . . . . . . 14 (𝜑 → (0g𝑅) ∈ V)
4740, 42, 45, 46suppssr 8127 . . . . . . . . . . . . 13 ((𝜑𝑒 ∈ ({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∖ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑀})) → (𝑃𝑒) = (0g𝑅))
4828, 38, 47syl2anc 584 . . . . . . . . . . . 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 7372 . . . . . . . . . . 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 730 . . . . . . . . . . . 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 730 . . . . . . . . . . . . . 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 21404 . . . . . . . . . . . . 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 782 . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . 16 {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} = {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}
555, 54psrbagconcl 21336 . . . . . . . . . . . . . . 15 ((𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑥f𝑒) ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥})
5653, 32, 55syl2anc 584 . . . . . . . . . . . . . 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 3639 . . . . . . . . . . . . . 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, 58ffvelcdmd 7036 . . . . . . . . . . . 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 20011 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑄‘(𝑥f𝑒)) ∈ (Base‘𝑅)) → ((0g𝑅)(.r𝑅)(𝑄‘(𝑥f𝑒))) = (0g𝑅))
6150, 59, 60syl2anc 584 . . . . . . . . . . 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 2776 . . . . . . . . . 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 781 . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . . . 17 (𝑐 = (𝑥f𝑒) → ((ℂflds0) Σg 𝑐) = ((ℂflds0) Σg (𝑥f𝑒)))
6564eqeq1d 2738 . . . . . . . . . . . . . . . 16 (𝑐 = (𝑥f𝑒) → (((ℂflds0) Σg 𝑐) = 𝑁 ↔ ((ℂflds0) Σg (𝑥f𝑒)) = 𝑁))
6665necon3bbid 2981 . . . . . . . . . . . . . . 15 (𝑐 = (𝑥f𝑒) → (¬ ((ℂflds0) Σg 𝑐) = 𝑁 ↔ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁))
67 simp-4r 782 . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . 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 3647 . . . . . . . . . . . . . 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 4271 . . . . . . . . . . . . . 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 21404 . . . . . . . . . . . . . 14 (𝜑𝑄:{ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}⟶(Base‘𝑅))
766, 41, 5, 7, 8, 12, 13mhpdeg 21535 . . . . . . . . . . . . . 14 (𝜑 → (𝑄 supp (0g𝑅)) ⊆ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑁})
7775, 76, 45, 46suppssr 8127 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥f𝑒) ∈ ({ ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∖ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ ((ℂflds0) Σg 𝑐) = 𝑁})) → (𝑄‘(𝑥f𝑒)) = (0g𝑅))
7863, 74, 77syl2anc 584 . . . . . . . . . . . 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 7373 . . . . . . . . . . 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 730 . . . . . . . . . . . 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 730 . . . . . . . . . . . . . 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 21404 . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . 14 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑒 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
8483adantr 481 . . . . . . . . . . . . 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, 84ffvelcdmd 7036 . . . . . . . . . . . 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 20012 . . . . . . . . . . . 12 ((𝑅 ∈ Ring ∧ (𝑃𝑒) ∈ (Base‘𝑅)) → ((𝑃𝑒)(.r𝑅)(0g𝑅)) = (0g𝑅))
8780, 85, 86syl2anc 584 . . . . . . . . . . 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 2776 . . . . . . . . . 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 20852 . . . . . . . . . . . . . . . 16 0 ∈ (SubMnd‘ℂfld)
90 eqid 2736 . . . . . . . . . . . . . . . . 17 (ℂflds0) = (ℂflds0)
9190submbas 18625 . . . . . . . . . . . . . . . 16 (ℕ0 ∈ (SubMnd‘ℂfld) → ℕ0 = (Base‘(ℂflds0)))
9289, 91ax-mp 5 . . . . . . . . . . . . . . 15 0 = (Base‘(ℂflds0))
93 cnfld0 20821 . . . . . . . . . . . . . . . . 17 0 = (0g‘ℂfld)
9490, 93subm0 18626 . . . . . . . . . . . . . . . 16 (ℕ0 ∈ (SubMnd‘ℂfld) → 0 = (0g‘(ℂflds0)))
9589, 94ax-mp 5 . . . . . . . . . . . . . . 15 0 = (0g‘(ℂflds0))
96 nn0ex 12419 . . . . . . . . . . . . . . . 16 0 ∈ V
97 cnfldadd 20801 . . . . . . . . . . . . . . . . 17 + = (+g‘ℂfld)
9890, 97ressplusg 17171 . . . . . . . . . . . . . . . 16 (ℕ0 ∈ V → + = (+g‘(ℂflds0)))
9996, 98ax-mp 5 . . . . . . . . . . . . . . 15 + = (+g‘(ℂflds0))
100 cnring 20819 . . . . . . . . . . . . . . . . . 18 fld ∈ Ring
101 ringcmn 20003 . . . . . . . . . . . . . . . . . 18 (ℂfld ∈ Ring → ℂfld ∈ CMnd)
102100, 101ax-mp 5 . . . . . . . . . . . . . . . . 17 fld ∈ CMnd
10390submcmn 19616 . . . . . . . . . . . . . . . . 17 ((ℂfld ∈ CMnd ∧ ℕ0 ∈ (SubMnd‘ℂfld)) → (ℂflds0) ∈ CMnd)
104102, 89, 103mp2an 690 . . . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝐼𝑉)
1075psrbagf 21320 . . . . . . . . . . . . . . . 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 774 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin})
1105psrbagf 21320 . . . . . . . . . . . . . . . . . . 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 6669 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑥 Fn 𝐼)
113108ffnd 6669 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑒 Fn 𝐼)
114 inidm 4178 . . . . . . . . . . . . . . . . 17 (𝐼𝐼) = 𝐼
115 eqidd 2737 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → (𝑥𝑖) = (𝑥𝑖))
116 eqidd 2737 . . . . . . . . . . . . . . . . 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 7626 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑥f𝑒) = (𝑖𝐼 ↦ ((𝑥𝑖) − (𝑒𝑖))))
118 simpl 483 . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . 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 5108 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = 𝑒 → (𝑐r𝑥𝑒r𝑥))
121120elrab 3645 . . . . . . . . . . . . . . . . . . . 20 (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↔ (𝑒 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∧ 𝑒r𝑥))
122121simprbi 497 . . . . . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . . . . . . 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 7629 . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → (𝑒𝑖) ≤ (𝑥𝑖))
127108ffvelcdmda 7035 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → (𝑒𝑖) ∈ ℕ0)
128111ffvelcdmda 7035 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → (𝑥𝑖) ∈ ℕ0)
129 nn0sub 12463 . . . . . . . . . . . . . . . . . 18 (((𝑒𝑖) ∈ ℕ0 ∧ (𝑥𝑖) ∈ ℕ0) → ((𝑒𝑖) ≤ (𝑥𝑖) ↔ ((𝑥𝑖) − (𝑒𝑖)) ∈ ℕ0))
130127, 128, 129syl2anc 584 . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑖𝐼) → ((𝑥𝑖) − (𝑒𝑖)) ∈ ℕ0)
132117, 131fmpt3d 7064 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑥f𝑒):𝐼⟶ℕ0)
133108ffund 6672 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → Fun 𝑒)
134 c0ex 11149 . . . . . . . . . . . . . . . . . . . 20 0 ∈ V
135106, 134jctir 521 . . . . . . . . . . . . . . . . . . 19 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝐼𝑉 ∧ 0 ∈ V))
136 fsuppeq 8106 . . . . . . . . . . . . . . . . . . 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 12426 . . . . . . . . . . . . . . . . . . 19 ℕ = (ℕ0 ∖ {0})
139138imaeq2i 6011 . . . . . . . . . . . . . . . . . 18 (𝑒 “ ℕ) = (𝑒 “ (ℕ0 ∖ {0}))
140137, 139eqtr4di 2794 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒 supp 0) = (𝑒 “ ℕ))
1415psrbag 21319 . . . . . . . . . . . . . . . . . . . 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 231 . . . . . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . . . . 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 3465 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑒 ∈ V)
147 isfsupp 9309 . . . . . . . . . . . . . . . . 17 ((𝑒 ∈ V ∧ 0 ∈ V) → (𝑒 finSupp 0 ↔ (Fun 𝑒 ∧ (𝑒 supp 0) ∈ Fin)))
148146, 134, 147sylancl 586 . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . 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 7631 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → Fun (𝑥f𝑒))
1515psrbagfsupp 21322 . . . . . . . . . . . . . . . . . . 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 9325 . . . . . . . . . . . . . . . . 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 12428 . . . . . . . . . . . . . . . . . . 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 12273 . . . . . . . . . . . . . . . . . . 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 8134 . . . . . . . . . . . . . . . . 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 9211 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((𝑥f𝑒) supp 0) ∈ Fin)
160 ovexd 7392 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑥f𝑒) ∈ V)
161 isfsupp 9309 . . . . . . . . . . . . . . . . 17 (((𝑥f𝑒) ∈ V ∧ 0 ∈ V) → ((𝑥f𝑒) finSupp 0 ↔ (Fun (𝑥f𝑒) ∧ ((𝑥f𝑒) supp 0) ∈ Fin)))
162160, 134, 161sylancl 586 . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . 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 19700 . . . . . . . . . . . . . 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))))
165108ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → (𝑒𝑏) ∈ ℕ0)
166165nn0cnd 12475 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → (𝑒𝑏) ∈ ℂ)
167111ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . 19 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → (𝑥𝑏) ∈ ℕ0)
168167nn0cnd 12475 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → (𝑥𝑏) ∈ ℂ)
169166, 168pncan3d 11515 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → ((𝑒𝑏) + ((𝑥𝑏) − (𝑒𝑏))) = (𝑥𝑏))
170169mpteq2dva 5205 . . . . . . . . . . . . . . . 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 6857 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → (𝑒𝑏) ∈ V)
172 ovexd 7392 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) ∧ 𝑏𝐼) → ((𝑥𝑏) − (𝑒𝑏)) ∈ V)
173108feqmptd 6910 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → 𝑒 = (𝑏𝐼 ↦ (𝑒𝑏)))
174111feqmptd 6910 . . . . . . . . . . . . . . . . . 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 7637 . . . . . . . . . . . . . . . . 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 7637 . . . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . . . 15 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → (𝑒f + (𝑥f𝑒)) = 𝑥)
178177oveq2d 7373 . . . . . . . . . . . . . 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 2778 . . . . . . . . . . . . 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 767 . . . . . . . . . . . . 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 3011 . . . . . . . . . . . 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 7366 . . . . . . . . . . . . . 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 2956 . . . . . . . . . . . 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 3039 . . . . . . . . . . 11 ((((ℂflds0) Σg 𝑒) ≠ 𝑀 ∨ ((ℂflds0) Σg (𝑥f𝑒)) ≠ 𝑁) ↔ ¬ (((ℂflds0) Σg 𝑒) = 𝑀 ∧ ((ℂflds0) Σg (𝑥f𝑒)) = 𝑁))
187185, 186sylibr 233 . . . . . . . . . 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 957 . . . . . . . . 9 ((((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) ∧ 𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥}) → ((𝑃𝑒)(.r𝑅)(𝑄‘(𝑥f𝑒))) = (0g𝑅))
189188mpteq2dva 5205 . . . . . . . 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 7373 . . . . . . 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 19974 . . . . . . . . . 10 (𝑅 ∈ Ring → 𝑅 ∈ Mnd)
1928, 191syl 17 . . . . . . . . 9 (𝜑𝑅 ∈ Mnd)
193192ad2antrr 724 . . . . . . . 8 (((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → 𝑅 ∈ Mnd)
19444rabex 5289 . . . . . . . 8 {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ∈ V
19541gsumz 18646 . . . . . . . 8 ((𝑅 ∈ Mnd ∧ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ∈ V) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ (0g𝑅))) = (0g𝑅))
196193, 194, 195sylancl 586 . . . . . . 7 (((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) ∧ ((ℂflds0) Σg 𝑥) ≠ (𝑀 + 𝑁)) → (𝑅 Σg (𝑒 ∈ {𝑐 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} ∣ 𝑐r𝑥} ↦ (0g𝑅))) = (0g𝑅))
197190, 196eqtrd 2776 . . . . . 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 413 . . . . 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 2965 . . . 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 239 . . 3 ((𝜑𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin}) → (((𝑃 · 𝑄)‘𝑥) ≠ (0g𝑅) → ((ℂflds0) Σg 𝑥) = (𝑀 + 𝑁)))
201200ralrimiva 3143 . 2 (𝜑 → ∀𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} (((𝑃 · 𝑄)‘𝑥) ≠ (0g𝑅) → ((ℂflds0) Σg 𝑥) = (𝑀 + 𝑁)))
2029, 12nn0addcld 12477 . . 3 (𝜑 → (𝑀 + 𝑁) ∈ ℕ0)
2031mplring 21424 . . . . 5 ((𝐼𝑉𝑅 ∈ Ring) → 𝑌 ∈ Ring)
2047, 8, 203syl2anc 584 . . . 4 (𝜑𝑌 ∈ Ring)
2052, 4ringcl 19981 . . . 4 ((𝑌 ∈ Ring ∧ 𝑃 ∈ (Base‘𝑌) ∧ 𝑄 ∈ (Base‘𝑌)) → (𝑃 · 𝑄) ∈ (Base‘𝑌))
206204, 11, 14, 205syl3anc 1371 . . 3 (𝜑 → (𝑃 · 𝑄) ∈ (Base‘𝑌))
2076, 1, 2, 41, 5, 7, 8, 202, 206ismhp3 21533 . 2 (𝜑 → ((𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)) ↔ ∀𝑥 ∈ { ∈ (ℕ0m 𝐼) ∣ ( “ ℕ) ∈ Fin} (((𝑃 · 𝑄)‘𝑥) ≠ (0g𝑅) → ((ℂflds0) Σg 𝑥) = (𝑀 + 𝑁))))
208201, 207mpbird 256 1 (𝜑 → (𝑃 · 𝑄) ∈ (𝐻‘(𝑀 + 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  wne 2943  wral 3064  {crab 3407  Vcvv 3445  cdif 3907  cun 3908  {csn 4586   class class class wbr 5105  cmpt 5188  ccnv 5632  cima 5636  Fun wfun 6490  wf 6492  cfv 6496  (class class class)co 7357  f cof 7615  r cofr 7616   supp csupp 8092  m cmap 8765  Fincfn 8883   finSupp cfsupp 9305  0cc0 11051   + caddc 11054  cle 11190  cmin 11385  cn 12153  0cn0 12413  Basecbs 17083  s cress 17112  +gcplusg 17133  .rcmulr 17134  0gc0g 17321   Σg cgsu 17322  Mndcmnd 18556  SubMndcsubmnd 18600  CMndccmn 19562  Ringcrg 19964  fldccnfld 20796   mPoly cmpl 21308   mHomP cmhp 21519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-of 7617  df-ofr 7618  df-om 7803  df-1st 7921  df-2nd 7922  df-supp 8093  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-ixp 8836  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-fsupp 9306  df-sup 9378  df-oi 9446  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-fz 13425  df-fzo 13568  df-seq 13907  df-hash 14231  df-struct 17019  df-sets 17036  df-slot 17054  df-ndx 17066  df-base 17084  df-ress 17113  df-plusg 17146  df-mulr 17147  df-starv 17148  df-sca 17149  df-vsca 17150  df-ip 17151  df-tset 17152  df-ple 17153  df-ds 17155  df-unif 17156  df-hom 17157  df-cco 17158  df-0g 17323  df-gsum 17324  df-prds 17329  df-pws 17331  df-mre 17466  df-mrc 17467  df-acs 17469  df-mgm 18497  df-sgrp 18546  df-mnd 18557  df-mhm 18601  df-submnd 18602  df-grp 18751  df-minusg 18752  df-mulg 18873  df-subg 18925  df-ghm 19006  df-cntz 19097  df-cmn 19564  df-abl 19565  df-mgp 19897  df-ur 19914  df-ring 19966  df-cring 19967  df-subrg 20220  df-cnfld 20797  df-psr 21311  df-mpl 21313  df-mhp 21523
This theorem is referenced by:  mhppwdeg  21540
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