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Theorem breprexplema 32077
 Description: Lemma for breprexp 32080 (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 12953 . . . . 5 (1...𝑁) ⊆ ℕ
21a1i 11 . . . 4 (𝜑 → (1...𝑁) ⊆ ℕ)
3 breprexplema.m . . . . 5 (𝜑𝑀 ∈ ℕ0)
43nn0zd 12093 . . . 4 (𝜑𝑀 ∈ ℤ)
5 breprexp.s . . . 4 (𝜑𝑆 ∈ ℕ0)
6 eqid 2798 . . . 4 (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) = (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
72, 4, 5, 6reprsuc 32062 . . 3 (𝜑 → ((1...𝑁)(repr‘(𝑆 + 1))𝑀) = 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
87sumeq1d 15070 . 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 13356 . . 3 (𝜑 → (1...𝑁) ∈ Fin)
101a1i 11 . . . . . 6 ((𝜑𝑏 ∈ (1...𝑁)) → (1...𝑁) ⊆ ℕ)
114adantr 484 . . . . . . 7 ((𝜑𝑏 ∈ (1...𝑁)) → 𝑀 ∈ ℤ)
12 fzssz 12924 . . . . . . . 8 (1...𝑁) ⊆ ℤ
13 simpr 488 . . . . . . . 8 ((𝜑𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
1412, 13sseldi 3915 . . . . . . 7 ((𝜑𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℤ)
1511, 14zsubcld 12100 . . . . . 6 ((𝜑𝑏 ∈ (1...𝑁)) → (𝑀𝑏) ∈ ℤ)
165adantr 484 . . . . . 6 ((𝜑𝑏 ∈ (1...𝑁)) → 𝑆 ∈ ℕ0)
179adantr 484 . . . . . 6 ((𝜑𝑏 ∈ (1...𝑁)) → (1...𝑁) ∈ Fin)
1810, 15, 16, 17reprfi 32063 . . . . 5 ((𝜑𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ∈ Fin)
19 mptfi 8825 . . . . 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 8809 . . . 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 32057 . . . . 5 ((𝜑𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) = {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = (𝑀𝑏)})
24 ssrab2 4009 . . . . 5 {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐𝑎) = (𝑀𝑏)} ⊆ ((1...𝑁) ↑m (0..^𝑆))
2523, 24eqsstrdi 3971 . . . 4 ((𝜑𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ⊆ ((1...𝑁) ↑m (0..^𝑆)))
269elexd 3462 . . . 4 (𝜑 → (1...𝑁) ∈ V)
27 fzonel 13066 . . . . 5 ¬ 𝑆 ∈ (0..^𝑆)
2827a1i 11 . . . 4 (𝜑 → ¬ 𝑆 ∈ (0..^𝑆))
2925, 26, 5, 28, 6actfunsnrndisj 32052 . . 3 (𝜑Disj 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
30 fzofi 13357 . . . . . 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 3149 . . . . . . . 8 ((𝜑𝑥 ∈ (0..^(𝑆 + 1))) → ∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ)
3433ralrimiva 3149 . . . . . . 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 488 . . . . . . 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 1915 . . . . . . . . . . . 12 𝑣(𝜑𝑏 ∈ (1...𝑁))
38 nfcv 2955 . . . . . . . . . . . . 13 𝑣𝑑
39 nfmpt1 5132 . . . . . . . . . . . . . 14 𝑣(𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
4039nfrn 5792 . . . . . . . . . . . . 13 𝑣ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
4138, 40nfel 2969 . . . . . . . . . . . 12 𝑣 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
4237, 41nfan 1900 . . . . . . . . . . 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 32059 . . . . . . . . . . . . 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 6641 . . . . . . . . . . . . 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 13090 . . . . . . . . . . . . . 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 6523 . . . . . . . . . . . . 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 836 . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 12288 . . . . . . . . . . . . . . . 16 0 = (ℤ‘0)
565, 55eleqtrdi 2900 . . . . . . . . . . . . . . 15 (𝜑𝑆 ∈ (ℤ‘0))
57 fzosplitsn 13160 . . . . . . . . . . . . . . 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 6483 . . . . . . . . . . . 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 260 . . . . . . . . . . 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 488 . . . . . . . . . . . 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 3445 . . . . . . . . . . . . . 14 𝑣 ∈ V
64 snex 5301 . . . . . . . . . . . . . 14 {⟨𝑆, 𝑏⟩} ∈ V
6563, 64unex 7462 . . . . . . . . . . . . 13 (𝑣 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
666, 65elrnmpti 5800 . . . . . . . . . . . 12 (𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ↔ ∃𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
6762, 66sylib 221 . . . . . . . . . . 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 3290 . . . . . . . . . 10 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → 𝑑:(0..^(𝑆 + 1))⟶(1...𝑁))
6968adantr 484 . . . . . . . . 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, 36ffvelrnd 6839 . . . . . . . 8 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑𝑎) ∈ (1...𝑁))
711, 70sseldi 3915 . . . . . . 7 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑𝑎) ∈ ℕ)
72 fveq2 6655 . . . . . . . . . 10 (𝑥 = 𝑎 → (𝐿𝑥) = (𝐿𝑎))
7372fveq1d 6657 . . . . . . . . 9 (𝑥 = 𝑎 → ((𝐿𝑥)‘𝑦) = ((𝐿𝑎)‘𝑦))
7473eleq1d 2874 . . . . . . . 8 (𝑥 = 𝑎 → (((𝐿𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿𝑎)‘𝑦) ∈ ℂ))
75 fveq2 6655 . . . . . . . . 9 (𝑦 = (𝑑𝑎) → ((𝐿𝑎)‘𝑦) = ((𝐿𝑎)‘(𝑑𝑎)))
7675eleq1d 2874 . . . . . . . 8 (𝑦 = (𝑑𝑎) → (((𝐿𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿𝑎)‘(𝑑𝑎)) ∈ ℂ))
7774, 76rspc2v 3582 . . . . . . 7 ((𝑎 ∈ (0..^(𝑆 + 1)) ∧ (𝑑𝑎) ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ → ((𝐿𝑎)‘(𝑑𝑎)) ∈ ℂ))
7836, 71, 77syl2anc 587 . . . . . 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 15318 . . . 4 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿𝑎)‘(𝑑𝑎)) ∈ ℂ)
8180anasss 470 . . 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 15188 . 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 15287 . . . . . 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 1915 . . . . . . 7 𝑎((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)))
86 nfcv 2955 . . . . . . 7 𝑎((𝐿𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
87 fzofi 13357 . . . . . . . 8 (0..^𝑆) ∈ Fin
8887a1i 11 . . . . . . 7 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (0..^𝑆) ∈ Fin)
8916adantr 484 . . . . . . 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 484 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → (𝑀𝑏) ∈ ℤ)
93 simpr 488 . . . . . . . . . . . . 13 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏)))
9491, 92, 89, 93reprf 32059 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑒:(0..^𝑆)⟶(1...𝑁))
9594ffnd 6496 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑒 Fn (0..^𝑆))
9695adantr 484 . . . . . . . . . 10 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒 Fn (0..^𝑆))
9713adantr 484 . . . . . . . . . . . 12 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑏 ∈ (1...𝑁))
98 fnsng 6384 . . . . . . . . . . . 12 ((𝑆 ∈ ℕ0𝑏 ∈ (1...𝑁)) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
9989, 97, 98syl2anc 587 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
10099adantr 484 . . . . . . . . . 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 488 . . . . . . . . . 10 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
103 fvun1 6739 . . . . . . . . . 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 1371 . . . . . . . . 9 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑒𝑎))
105104fveq2d 6659 . . . . . . . 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 484 . . . . . . . . 9 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ)
108 fzossfzop1 13130 . . . . . . . . . . . . 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 3917 . . . . . . . . . 10 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
11294ffvelrnda 6838 . . . . . . . . . . 11 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒𝑎) ∈ (1...𝑁))
1131, 112sseldi 3915 . . . . . . . . . 10 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒𝑎) ∈ ℕ)
114 fveq2 6655 . . . . . . . . . . . 12 (𝑦 = (𝑒𝑎) → ((𝐿𝑎)‘𝑦) = ((𝐿𝑎)‘(𝑒𝑎)))
115114eleq1d 2874 . . . . . . . . . . 11 (𝑦 = (𝑒𝑎) → (((𝐿𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿𝑎)‘(𝑒𝑎)) ∈ ℂ))
11674, 115rspc2v 3582 . . . . . . . . . 10 ((𝑎 ∈ (0..^(𝑆 + 1)) ∧ (𝑒𝑎) ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ → ((𝐿𝑎)‘(𝑒𝑎)) ∈ ℂ))
117111, 113, 116syl2anc 587 . . . . . . . . 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 2890 . . . . . . 7 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) ∈ ℂ)
120 fveq2 6655 . . . . . . . 8 (𝑎 = 𝑆 → (𝐿𝑎) = (𝐿𝑆))
121 fveq2 6655 . . . . . . . 8 (𝑎 = 𝑆 → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
122120, 121fveq12d 6662 . . . . . . 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 4562 . . . . . . . . . . . 12 (𝑆 ∈ ℕ0𝑆 ∈ {𝑆})
12589, 124syl 17 . . . . . . . . . . 11 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑆 ∈ {𝑆})
126 fvun2 6740 . . . . . . . . . . 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 1371 . . . . . . . . . 10 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
128 fvsng 6929 . . . . . . . . . . 11 ((𝑆 ∈ ℕ0𝑏 ∈ (1...𝑁)) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
12989, 97, 128syl2anc 587 . . . . . . . . . 10 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
130127, 129eqtrd 2833 . . . . . . . . 9 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = 𝑏)
131130fveq2d 6659 . . . . . . . 8 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((𝐿𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) = ((𝐿𝑆)‘𝑏))
132 fzonn0p1 13129 . . . . . . . . . . . 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, 97sseldi 3915 . . . . . . . . . 10 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → 𝑏 ∈ ℕ)
136 fveq2 6655 . . . . . . . . . . . . 13 (𝑥 = 𝑆 → (𝐿𝑥) = (𝐿𝑆))
137136fveq1d 6657 . . . . . . . . . . . 12 (𝑥 = 𝑆 → ((𝐿𝑥)‘𝑦) = ((𝐿𝑆)‘𝑦))
138137eleq1d 2874 . . . . . . . . . . 11 (𝑥 = 𝑆 → (((𝐿𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿𝑆)‘𝑦) ∈ ℂ))
139 fveq2 6655 . . . . . . . . . . . 12 (𝑦 = 𝑏 → ((𝐿𝑆)‘𝑦) = ((𝐿𝑆)‘𝑏))
140139eleq1d 2874 . . . . . . . . . . 11 (𝑦 = 𝑏 → (((𝐿𝑆)‘𝑦) ∈ ℂ ↔ ((𝐿𝑆)‘𝑏) ∈ ℂ))
141138, 140rspc2v 3582 . . . . . . . . . 10 ((𝑆 ∈ (0..^(𝑆 + 1)) ∧ 𝑏 ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿𝑥)‘𝑦) ∈ ℂ → ((𝐿𝑆)‘𝑏) ∈ ℂ))
142134, 135, 141syl2anc 587 . . . . . . . . 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 2890 . . . . . . 7 (((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) → ((𝐿𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) ∈ ℂ)
14585, 86, 88, 89, 90, 119, 122, 144fprodsplitsn 15355 . . . . . 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 15289 . . . . . . 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 7163 . . . . . 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 2837 . . . . 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 15072 . . . 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 486 . . . . . . . 8 ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → 𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
151150fveq1d 6657 . . . . . . 7 ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎))
152151fveq2d 6659 . . . . . 6 ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ((𝐿𝑎)‘(𝑑𝑎)) = ((𝐿𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
153152prodeq2dv 15289 . . . . 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 32051 . . . . 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 488 . . . . . . 7 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑣 = 𝑒) → 𝑣 = 𝑒)
157156uneq1d 4092 . . . . . 6 ((((𝜑𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀𝑏))) ∧ 𝑣 = 𝑒) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}) = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
158 vex 3445 . . . . . . . 8 𝑒 ∈ V
159158, 64unex 7462 . . . . . . 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 6762 . . . . 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 15092 . . . 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 486 . . . . . . . . . 10 ((𝑑 = 𝑒𝑎 ∈ (0..^𝑆)) → 𝑑 = 𝑒)
164163fveq1d 6657 . . . . . . . . 9 ((𝑑 = 𝑒𝑎 ∈ (0..^𝑆)) → (𝑑𝑎) = (𝑒𝑎))
165164fveq2d 6659 . . . . . . . 8 ((𝑑 = 𝑒𝑎 ∈ (0..^𝑆)) → ((𝐿𝑎)‘(𝑑𝑎)) = ((𝐿𝑎)‘(𝑒𝑎)))
166165prodeq2dv 15289 . . . . . . 7 (𝑑 = 𝑒 → ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑒𝑎)))
167166oveq1d 7160 . . . . . 6 (𝑑 = 𝑒 → (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑑𝑎)) · ((𝐿𝑆)‘𝑏)) = (∏𝑎 ∈ (0..^𝑆)((𝐿𝑎)‘(𝑒𝑎)) · ((𝐿𝑆)‘𝑏)))
168167cbvsumv 15065 . . . . 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 2843 . . 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 15072 . 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 2837 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 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3442   ∪ cun 3881   ∩ cin 3882   ⊆ wss 3883  ∅c0 4246  {csn 4528  ⟨cop 4534  ∪ ciun 4885   class class class wbr 5034   ↦ cmpt 5114  ran crn 5524   Fn wfn 6327  ⟶wf 6328  ‘cfv 6332  (class class class)co 7145   ↑m cmap 8407  Fincfn 8510  ℂcc 10542  0cc0 10544  1c1 10545   + caddc 10547   · cmul 10549   ≤ cle 10683   − cmin 10877  ℕcn 11643  ℕ0cn0 11903  ℤcz 11989  ℤ≥cuz 12251  ...cfz 12905  ..^cfzo 13048  Σcsu 15054  ∏cprod 15271  reprcrepr 32055 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-inf2 9106  ax-cnex 10600  ax-resscn 10601  ax-1cn 10602  ax-icn 10603  ax-addcl 10604  ax-addrcl 10605  ax-mulcl 10606  ax-mulrcl 10607  ax-mulcom 10608  ax-addass 10609  ax-mulass 10610  ax-distr 10611  ax-i2m1 10612  ax-1ne0 10613  ax-1rid 10614  ax-rnegex 10615  ax-rrecex 10616  ax-cnre 10617  ax-pre-lttri 10618  ax-pre-lttrn 10619  ax-pre-ltadd 10620  ax-pre-mulgt0 10621  ax-pre-sup 10622 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4805  df-int 4843  df-iun 4887  df-disj 5000  df-br 5035  df-opab 5097  df-mpt 5115  df-tr 5141  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-isom 6341  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7574  df-1st 7684  df-2nd 7685  df-wrecs 7948  df-recs 8009  df-rdg 8047  df-1o 8103  df-2o 8104  df-oadd 8107  df-er 8290  df-map 8409  df-pm 8410  df-en 8511  df-dom 8512  df-sdom 8513  df-fin 8514  df-sup 8908  df-oi 8976  df-card 9370  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688  df-sub 10879  df-neg 10880  df-div 11305  df-nn 11644  df-2 11706  df-3 11707  df-n0 11904  df-z 11990  df-uz 12252  df-rp 12398  df-fz 12906  df-fzo 13049  df-seq 13385  df-exp 13446  df-hash 13707  df-cj 14470  df-re 14471  df-im 14472  df-sqrt 14606  df-abs 14607  df-clim 14857  df-sum 15055  df-prod 15272  df-repr 32056 This theorem is referenced by:  breprexplemc  32079
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