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Theorem breprexplema 34262
Description: Lemma for breprexp 34265 (induction step for weighted sums over representations). (Contributed by Thierry Arnoux, 7-Dec-2021.)
Hypotheses
Ref Expression
breprexp.n (𝜑𝑁 ∈ ℕ0)
breprexp.s (𝜑𝑆 ∈ ℕ0)
breprexplema.m (𝜑𝑀 ∈ ℕ0)
breprexplema.1 (𝜑𝑀 ≤ ((𝑆 + 1) · 𝑁))
breprexplema.l (((𝜑𝑥 ∈ (0..^(𝑆 + 1))) ∧ 𝑦 ∈ ℕ) → ((𝐿𝑥)‘𝑦) ∈ ℂ)
Assertion
Ref Expression
breprexplema (𝜑 → Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑆 + 1))𝑀)∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) = Σ𝑏 ∈ (1...𝑁𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) · ((𝐿𝑆)‘𝑏)))
Distinct variable groups:   𝑆,𝑎   𝐿,𝑎,𝑏,𝑑,𝑥,𝑦   𝑀,𝑎,𝑏,𝑑   𝑁,𝑎,𝑏,𝑑   𝑆,𝑏,𝑑,𝑥,𝑦   𝜑,𝑎,𝑏,𝑑,𝑥,𝑦
Allowed substitution hints:   𝑀(𝑥,𝑦)   𝑁(𝑥,𝑦)

Proof of Theorem breprexplema
Dummy variables 𝑐 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fz1ssnn 13564 . . . . 5 (1...𝑁) ⊆ ℕ
21a1i 11 . . . 4 (𝜑 → (1...𝑁) ⊆ ℕ)
3 breprexplema.m . . . . 5 (𝜑𝑀 ∈ ℕ0)
43nn0zd 12614 . . . 4 (𝜑𝑀 ∈ ℤ)
5 breprexp.s . . . 4 (𝜑𝑆 ∈ ℕ0)
6 eqid 2728 . . . 4 (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) = (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
72, 4, 5, 6reprsuc 34247 . . 3 (𝜑 → ((1...𝑁)(repr‘(𝑆 + 1))𝑀) = 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
87sumeq1d 15679 . 2 (𝜑 → Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑆 + 1))𝑀)∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) = Σ𝑑 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)))
9 fzfid 13970 . . 3 (𝜑 → (1...𝑁) ∈ Fin)
101a1i 11 . . . . . 6 ((𝜑𝑏 ∈ (1...𝑁)) → (1...𝑁) ⊆ ℕ)
114adantr 480 . . . . . . 7 ((𝜑𝑏 ∈ (1...𝑁)) → 𝑀 ∈ ℤ)
12 fzssz 13535 . . . . . . . 8 (1...𝑁) ⊆ ℤ
13 simpr 484 . . . . . . . 8 ((𝜑𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
1412, 13sselid 3978 . . . . . . 7 ((𝜑𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℤ)
1511, 14zsubcld 12701 . . . . . 6 ((𝜑𝑏 ∈ (1...𝑁)) → (𝑀𝑏) ∈ ℤ)
165adantr 480 . . . . . 6 ((𝜑𝑏 ∈ (1...𝑁)) → 𝑆 ∈ ℕ0)
179adantr 480 . . . . . 6 ((𝜑𝑏 ∈ (1...𝑁)) → (1...𝑁) ∈ Fin)
1810, 15, 16, 17reprfi 34248 . . . . 5 ((𝜑𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ∈ Fin)
19 mptfi 9375 . . . . 5 (((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ∈ Fin → (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
2018, 19syl 17 . . . 4 ((𝜑𝑏 ∈ (1...𝑁)) → (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
21 rnfi 9359 . . . 4 ((𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin → ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
2220, 21syl 17 . . 3 ((𝜑𝑏 ∈ (1...𝑁)) → ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
2310, 15, 16reprval 34242 . . . . 5 ((𝜑𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) = {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = (𝑀𝑏)})
24 ssrab2 4075 . . . . 5 {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = (𝑀𝑏)} ⊆ ((1...𝑁) ↑m (0..^𝑆))
2523, 24eqsstrdi 4034 . . . 4 ((𝜑𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ⊆ ((1...𝑁) ↑m (0..^𝑆)))
269elexd 3492 . . . 4 (𝜑 → (1...𝑁) ∈ V)
27 fzonel 13678 . . . . 5 ¬ 𝑆 ∈ (0..^𝑆)
2827a1i 11 . . . 4 (𝜑 → ¬ 𝑆 ∈ (0..^𝑆))
2925, 26, 5, 28, 6actfunsnrndisj 34237 . . 3 (𝜑Disj 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
30 fzofi 13971 . . . . . 6 (0..^(𝑆 + 1)) ∈ Fin
3130a1i 11 . . . . 5 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → (0..^(𝑆 + 1)) ∈ Fin)
32 breprexplema.l . . . . . . . . 9 (((𝜑𝑥 ∈ (0..^(𝑆 + 1))) ∧ 𝑦 ∈ ℕ) → ((𝐿𝑥)‘𝑦) ∈ ℂ)
3332ralrimiva 3143 . . . . . . . 8 ((𝜑𝑥 ∈ (0..^(𝑆 + 1))) → ∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ)
3433ralrimiva 3143 . . . . . . 7 (𝜑 → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ)
3534ad3antrrr 729 . . . . . 6 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ)
36 simpr 484 . . . . . . 7 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → 𝑎 ∈ (0..^(𝑆 + 1)))
37 nfv 1910 . . . . . . . . . . . 12 𝑣(𝜑𝑏 ∈ (1...𝑁))
38 nfcv 2899 . . . . . . . . . . . . 13 𝑣𝑑
39 nfmpt1 5256 . . . . . . . . . . . . . 14 𝑣(𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
4039nfrn 5954 . . . . . . . . . . . . 13 𝑣ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
4138, 40nfel 2914 . . . . . . . . . . . 12 𝑣 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
4237, 41nfan 1895 . . . . . . . . . . 11 𝑣((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
431a1i 11 . . . . . . . . . . . . . 14 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → (1...𝑁) ⊆ ℕ)
4415ad3antrrr 729 . . . . . . . . . . . . . 14 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → (𝑀𝑏) ∈ ℤ)
4516ad3antrrr 729 . . . . . . . . . . . . . 14 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑆 ∈ ℕ0)
46 simplr 768 . . . . . . . . . . . . . 14 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)))
4743, 44, 45, 46reprf 34244 . . . . . . . . . . . . 13 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑣:(0..^𝑆)⟶(1...𝑁))
4813ad3antrrr 729 . . . . . . . . . . . . . 14 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑏 ∈ (1...𝑁))
4945, 48fsnd 6882 . . . . . . . . . . . . 13 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶(1...𝑁))
50 fzodisjsn 13702 . . . . . . . . . . . . . 14 ((0..^𝑆) ∩ {𝑆}) = ∅
5150a1i 11 . . . . . . . . . . . . 13 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → ((0..^𝑆) ∩ {𝑆}) = ∅)
52 fun2 6760 . . . . . . . . . . . . 13 (((𝑣:(0..^𝑆)⟶(1...𝑁) ∧ {⟨𝑆, 𝑏⟩}:{𝑆}⟶(1...𝑁)) ∧ ((0..^𝑆) ∩ {𝑆}) = ∅) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}):((0..^𝑆) ∪ {𝑆})⟶(1...𝑁))
5347, 49, 51, 52syl21anc 837 . . . . . . . . . . . 12 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}):((0..^𝑆) ∪ {𝑆})⟶(1...𝑁))
54 simpr 484 . . . . . . . . . . . . 13 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
55 nn0uz 12894 . . . . . . . . . . . . . . . 16 0 = (ℤ‘0)
565, 55eleqtrdi 2839 . . . . . . . . . . . . . . 15 (𝜑𝑆 ∈ (ℤ‘0))
57 fzosplitsn 13772 . . . . . . . . . . . . . . 15 (𝑆 ∈ (ℤ‘0) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
5856, 57syl 17 . . . . . . . . . . . . . 14 (𝜑 → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
5958ad4antr 731 . . . . . . . . . . . . 13 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
6054, 59feq12d 6710 . . . . . . . . . . . 12 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → (𝑑:(0..^(𝑆 + 1))⟶(1...𝑁) ↔ (𝑣 ∪ {⟨𝑆, 𝑏⟩}):((0..^𝑆) ∪ {𝑆})⟶(1...𝑁)))
6153, 60mpbird 257 . . . . . . . . . . 11 (((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑑:(0..^(𝑆 + 1))⟶(1...𝑁))
62 simpr 484 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
63 vex 3475 . . . . . . . . . . . . . 14 𝑣 ∈ V
64 snex 5433 . . . . . . . . . . . . . 14 {⟨𝑆, 𝑏⟩} ∈ V
6563, 64unex 7748 . . . . . . . . . . . . 13 (𝑣 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
666, 65elrnmpti 5962 . . . . . . . . . . . 12 (𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ↔ ∃𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
6762, 66sylib 217 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → ∃𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
6842, 61, 67r19.29af 3262 . . . . . . . . . 10 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → 𝑑:(0..^(𝑆 + 1))⟶(1...𝑁))
6968adantr 480 . . . . . . . . 9 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → 𝑑:(0..^(𝑆 + 1))⟶(1...𝑁))
7069, 36ffvelcdmd 7095 . . . . . . . 8 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑𝑎) ∈ (1...𝑁))
711, 70sselid 3978 . . . . . . 7 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑𝑎) ∈ ℕ)
72 fveq2 6897 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝐿𝑥) = (𝐿𝑎))
7372fveq1d 6899 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝐿𝑥)‘𝑦) = ((𝐿𝑎)‘𝑦))
7473eleq1d 2814 . . . . . . . 8 (𝑥 = 𝑎 → (((𝐿𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿𝑎)‘𝑦) ∈ ℂ))
75 fveq2 6897 . . . . . . . . 9 (𝑦 = (𝑑𝑎) → ((𝐿𝑎)‘𝑦) = ((𝐿𝑎)‘(𝑑𝑎)))
7675eleq1d 2814 . . . . . . . 8 (𝑦 = (𝑑𝑎) → (((𝐿𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿𝑎)‘(𝑑𝑎)) ∈ ℂ))
7774, 76rspc2v 3620 . . . . . . 7 ((𝑎 ∈ (0..^(𝑆 + 1)) ∧ (𝑑𝑎) ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ → ((𝐿𝑎)‘(𝑑𝑎)) ∈ ℂ))
7836, 71, 77syl2anc 583 . . . . . 6 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ → ((𝐿𝑎)‘(𝑑𝑎)) ∈ ℂ))
7935, 78mpd 15 . . . . 5 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ((𝐿𝑎)‘(𝑑𝑎)) ∈ ℂ)
8031, 79fprodcl 15928 . . . 4 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) ∈ ℂ)
8180anasss 466 . . 3 ((𝜑 ∧ (𝑏 ∈ (1...𝑁) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) ∈ ℂ)
829, 22, 29, 81fsumiun 15799 . 2 (𝜑 → Σ𝑑 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) = Σ𝑏 ∈ (1...𝑁𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)))
8358ad2antrr 725 . . . . . . 7 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
8483prodeq1d 15897 . . . . . 6 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = ∏𝑎 ∈ ((0..^𝑆) ∪ {𝑆})((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
85 nfv 1910 . . . . . . 7 𝑎((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)))
86 nfcv 2899 . . . . . . 7 𝑎((𝐿𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
87 fzofi 13971 . . . . . . . 8 (0..^𝑆) ∈ Fin
8887a1i 11 . . . . . . 7 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (0..^𝑆) ∈ Fin)
8916adantr 480 . . . . . . 7 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑆 ∈ ℕ0)
9027a1i 11 . . . . . . 7 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ¬ 𝑆 ∈ (0..^𝑆))
911a1i 11 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (1...𝑁) ⊆ ℕ)
9215adantr 480 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (𝑀𝑏) ∈ ℤ)
93 simpr 484 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)))
9491, 92, 89, 93reprf 34244 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑒:(0..^𝑆)⟶(1...𝑁))
9594ffnd 6723 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑒 Fn (0..^𝑆))
9695adantr 480 . . . . . . . . . 10 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒 Fn (0..^𝑆))
9713adantr 480 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑏 ∈ (1...𝑁))
98 fnsng 6605 . . . . . . . . . . . 12 ((𝑆 ∈ ℕ0𝑏 ∈ (1...𝑁)) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
9989, 97, 98syl2anc 583 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
10099adantr 480 . . . . . . . . . 10 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
10150a1i 11 . . . . . . . . . 10 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((0..^𝑆) ∩ {𝑆}) = ∅)
102 simpr 484 . . . . . . . . . 10 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
103 fvun1 6989 . . . . . . . . . 10 ((𝑒 Fn (0..^𝑆) ∧ {⟨𝑆, 𝑏⟩} Fn {𝑆} ∧ (((0..^𝑆) ∩ {𝑆}) = ∅ ∧ 𝑎 ∈ (0..^𝑆))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑒𝑎))
10496, 100, 101, 102, 103syl112anc 1372 . . . . . . . . 9 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑒𝑎))
105104fveq2d 6901 . . . . . . . 8 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = ((𝐿𝑎)‘(𝑒𝑎)))
10634ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ)
107106adantr 480 . . . . . . . . 9 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ)
108 fzossfzop1 13742 . . . . . . . . . . . . 13 (𝑆 ∈ ℕ0 → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
1095, 108syl 17 . . . . . . . . . . . 12 (𝜑 → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
110109ad2antrr 725 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
111110sselda 3980 . . . . . . . . . 10 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
11294ffvelcdmda 7094 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒𝑎) ∈ (1...𝑁))
1131, 112sselid 3978 . . . . . . . . . 10 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒𝑎) ∈ ℕ)
114 fveq2 6897 . . . . . . . . . . . 12 (𝑦 = (𝑒𝑎) → ((𝐿𝑎)‘𝑦) = ((𝐿𝑎)‘(𝑒𝑎)))
115114eleq1d 2814 . . . . . . . . . . 11 (𝑦 = (𝑒𝑎) → (((𝐿𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿𝑎)‘(𝑒𝑎)) ∈ ℂ))
11674, 115rspc2v 3620 . . . . . . . . . 10 ((𝑎 ∈ (0..^(𝑆 + 1)) ∧ (𝑒𝑎) ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ → ((𝐿𝑎)‘(𝑒𝑎)) ∈ ℂ))
117111, 113, 116syl2anc 583 . . . . . . . . 9 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ → ((𝐿𝑎)‘(𝑒𝑎)) ∈ ℂ))
118107, 117mpd 15 . . . . . . . 8 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑒𝑎)) ∈ ℂ)
119105, 118eqeltrd 2829 . . . . . . 7 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) ∈ ℂ)
120 fveq2 6897 . . . . . . . 8 (𝑎 = 𝑆 → (𝐿𝑎) = (𝐿𝑆))
121 fveq2 6897 . . . . . . . 8 (𝑎 = 𝑆 → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
122120, 121fveq12d 6904 . . . . . . 7 (𝑎 = 𝑆 → ((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = ((𝐿𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)))
12350a1i 11 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((0..^𝑆) ∩ {𝑆}) = ∅)
124 snidg 4663 . . . . . . . . . . . 12 (𝑆 ∈ ℕ0𝑆 ∈ {𝑆})
12589, 124syl 17 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑆 ∈ {𝑆})
126 fvun2 6990 . . . . . . . . . . 11 ((𝑒 Fn (0..^𝑆) ∧ {⟨𝑆, 𝑏⟩} Fn {𝑆} ∧ (((0..^𝑆) ∩ {𝑆}) = ∅ ∧ 𝑆 ∈ {𝑆})) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
12795, 99, 123, 125, 126syl112anc 1372 . . . . . . . . . 10 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
128 fvsng 7189 . . . . . . . . . . 11 ((𝑆 ∈ ℕ0𝑏 ∈ (1...𝑁)) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
12989, 97, 128syl2anc 583 . . . . . . . . . 10 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
130127, 129eqtrd 2768 . . . . . . . . 9 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = 𝑏)
131130fveq2d 6901 . . . . . . . 8 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((𝐿𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) = ((𝐿𝑆)‘𝑏))
132 fzonn0p1 13741 . . . . . . . . . . . 12 (𝑆 ∈ ℕ0𝑆 ∈ (0..^(𝑆 + 1)))
1335, 132syl 17 . . . . . . . . . . 11 (𝜑𝑆 ∈ (0..^(𝑆 + 1)))
134133ad2antrr 725 . . . . . . . . . 10 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑆 ∈ (0..^(𝑆 + 1)))
1351, 97sselid 3978 . . . . . . . . . 10 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑏 ∈ ℕ)
136 fveq2 6897 . . . . . . . . . . . . 13 (𝑥 = 𝑆 → (𝐿𝑥) = (𝐿𝑆))
137136fveq1d 6899 . . . . . . . . . . . 12 (𝑥 = 𝑆 → ((𝐿𝑥)‘𝑦) = ((𝐿𝑆)‘𝑦))
138137eleq1d 2814 . . . . . . . . . . 11 (𝑥 = 𝑆 → (((𝐿𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿𝑆)‘𝑦) ∈ ℂ))
139 fveq2 6897 . . . . . . . . . . . 12 (𝑦 = 𝑏 → ((𝐿𝑆)‘𝑦) = ((𝐿𝑆)‘𝑏))
140139eleq1d 2814 . . . . . . . . . . 11 (𝑦 = 𝑏 → (((𝐿𝑆)‘𝑦) ∈ ℂ ↔ ((𝐿𝑆)‘𝑏) ∈ ℂ))
141138, 140rspc2v 3620 . . . . . . . . . 10 ((𝑆 ∈ (0..^(𝑆 + 1)) ∧ 𝑏 ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ → ((𝐿𝑆)‘𝑏) ∈ ℂ))
142134, 135, 141syl2anc 583 . . . . . . . . 9 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ → ((𝐿𝑆)‘𝑏) ∈ ℂ))
143106, 142mpd 15 . . . . . . . 8 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((𝐿𝑆)‘𝑏) ∈ ℂ)
144131, 143eqeltrd 2829 . . . . . . 7 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((𝐿𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) ∈ ℂ)
14585, 86, 88, 89, 90, 119, 122, 144fprodsplitsn 15965 . . . . . 6 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ∏𝑎 ∈ ((0..^𝑆) ∪ {𝑆})((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) · ((𝐿𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))))
146105prodeq2dv 15899 . . . . . . 7 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑒𝑎)))
147146, 131oveq12d 7438 . . . . . 6 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) · ((𝐿𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑒𝑎)) · ((𝐿𝑆)‘𝑏)))
14884, 145, 1473eqtrd 2772 . . . . 5 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑒𝑎)) · ((𝐿𝑆)‘𝑏)))
149148sumeq2dv 15681 . . . 4 ((𝜑𝑏 ∈ (1...𝑁)) → Σ𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = Σ𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑒𝑎)) · ((𝐿𝑆)‘𝑏)))
150 simpl 482 . . . . . . . 8 ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → 𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
151150fveq1d 6899 . . . . . . 7 ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎))
152151fveq2d 6901 . . . . . 6 ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ((𝐿𝑎)‘(𝑑𝑎)) = ((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
153152prodeq2dv 15899 . . . . 5 (𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) = ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
15425, 26, 5, 28, 6actfunsnf1o 34236 . . . . 5 ((𝜑𝑏 ∈ (1...𝑁)) → (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})):((1...𝑁)(repr‘𝑆)(𝑀𝑏))–1-1-onto→ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
1556a1i 11 . . . . . 6 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) = (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
156 simpr 484 . . . . . . 7 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑣 = 𝑒) → 𝑣 = 𝑒)
157156uneq1d 4161 . . . . . 6 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑣 = 𝑒) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}) = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
158 vex 3475 . . . . . . . 8 𝑒 ∈ V
159158, 64unex 7748 . . . . . . 7 (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
160159a1i 11 . . . . . 6 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∈ V)
161155, 157, 93, 160fvmptd 7012 . . . . 5 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))‘𝑒) = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
162153, 18, 154, 161, 80fsumf1o 15701 . . . 4 ((𝜑𝑏 ∈ (1...𝑁)) → Σ𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) = Σ𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
163 simpl 482 . . . . . . . . . 10 ((𝑑 = 𝑒𝑎 ∈ (0..^𝑆)) → 𝑑 = 𝑒)
164163fveq1d 6899 . . . . . . . . 9 ((𝑑 = 𝑒𝑎 ∈ (0..^𝑆)) → (𝑑𝑎) = (𝑒𝑎))
165164fveq2d 6901 . . . . . . . 8 ((𝑑 = 𝑒𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑑𝑎)) = ((𝐿𝑎)‘(𝑒𝑎)))
166165prodeq2dv 15899 . . . . . . 7 (𝑑 = 𝑒 → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑒𝑎)))
167166oveq1d 7435 . . . . . 6 (𝑑 = 𝑒 → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) · ((𝐿𝑆)‘𝑏)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑒𝑎)) · ((𝐿𝑆)‘𝑏)))
168167cbvsumv 15674 . . . . 5 Σ𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) · ((𝐿𝑆)‘𝑏)) = Σ𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑒𝑎)) · ((𝐿𝑆)‘𝑏))
169168a1i 11 . . . 4 ((𝜑𝑏 ∈ (1...𝑁)) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) · ((𝐿𝑆)‘𝑏)) = Σ𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑒𝑎)) · ((𝐿𝑆)‘𝑏)))
170149, 162, 1693eqtr4d 2778 . . 3 ((𝜑𝑏 ∈ (1...𝑁)) → Σ𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) · ((𝐿𝑆)‘𝑏)))
171170sumeq2dv 15681 . 2 (𝜑 → Σ𝑏 ∈ (1...𝑁𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) = Σ𝑏 ∈ (1...𝑁𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) · ((𝐿𝑆)‘𝑏)))
1728, 82, 1713eqtrd 2772 1 (𝜑 → Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑆 + 1))𝑀)∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) = Σ𝑏 ∈ (1...𝑁𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) · ((𝐿𝑆)‘𝑏)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395   = wceq 1534  wcel 2099  wral 3058  wrex 3067  {crab 3429  Vcvv 3471  cun 3945  cin 3946  wss 3947  c0 4323  {csn 4629  cop 4635   ciun 4996   class class class wbr 5148  cmpt 5231  ran crn 5679   Fn wfn 6543  wf 6544  cfv 6548  (class class class)co 7420  m cmap 8844  Fincfn 8963  cc 11136  0cc0 11138  1c1 11139   + caddc 11141   · cmul 11143  cle 11279  cmin 11474  cn 12242  0cn0 12502  cz 12588  cuz 12852  ...cfz 13516  ..^cfzo 13659  Σcsu 15664  cprod 15881  reprcrepr 34240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-inf2 9664  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215  ax-pre-sup 11216
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rmo 3373  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-1o 8486  df-er 8724  df-map 8846  df-pm 8847  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-sup 9465  df-oi 9533  df-card 9962  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-nn 12243  df-2 12305  df-3 12306  df-n0 12503  df-z 12589  df-uz 12853  df-rp 13007  df-fz 13517  df-fzo 13660  df-seq 13999  df-exp 14059  df-hash 14322  df-cj 15078  df-re 15079  df-im 15080  df-sqrt 15214  df-abs 15215  df-clim 15464  df-sum 15665  df-prod 15882  df-repr 34241
This theorem is referenced by:  breprexplemc  34264
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