Theorem breprexplema 34814

Description: Lemma for breprexp 34817 (induction step for weighted sums over representations). (Contributed by Thierry Arnoux, 7-Dec-2021.)

Hypotheses

Ref	Expression
breprexp.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
breprexp.s	⊢ (𝜑 → 𝑆 ∈ ℕ0)
breprexplema.m	⊢ (𝜑 → 𝑀 ∈ ℕ0)
breprexplema.1	⊢ (𝜑 → 𝑀 ≤ ((𝑆 + 1) · 𝑁))
breprexplema.l	⊢ (((𝜑 ∧ 𝑥 ∈ (0..^(𝑆 + 1))) ∧ 𝑦 ∈ ℕ) → ((𝐿‘𝑥)‘𝑦) ∈ ℂ)

Assertion

Ref	Expression
breprexplema	⊢ (𝜑 → Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑆 + 1))𝑀)∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑆)‘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.

Step	Hyp	Ref	Expression
1		fz1ssnn 13485	. . . . 5 ⊢ (1...𝑁) ⊆ ℕ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
3		breprexplema.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
4	3	nn0zd 12527	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		breprexp.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
6		eqid 2737	. . . 4 ⊢ (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) = (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
7	2, 4, 5, 6	reprsuc 34799	. . 3 ⊢ (𝜑 → ((1...𝑁)(repr‘(𝑆 + 1))𝑀) = ∪ 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
8	7	sumeq1d 15637	. 2 ⊢ (𝜑 → Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑆 + 1))𝑀)∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑑 ∈ ∪ 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)))
9		fzfid 13910	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
10	1	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (1...𝑁) ⊆ ℕ)
11	4	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑀 ∈ ℤ)
12		fzssz 13456	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℤ
13		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
14	12, 13	sselid 3933	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℤ)
15	11, 14	zsubcld 12615	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (𝑀 − 𝑏) ∈ ℤ)
16	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑆 ∈ ℕ0)
17	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (1...𝑁) ∈ Fin)
18	10, 15, 16, 17	reprfi 34800	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ∈ Fin)
19		mptfi 9265	. . . . 5 ⊢ (((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ∈ Fin → (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
20	18, 19	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
21		rnfi 9254	. . . 4 ⊢ ((𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin → ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
22	20, 21	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
23	10, 15, 16	reprval 34794	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) = {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = (𝑀 − 𝑏)})
24		ssrab2 4034	. . . . 5 ⊢ {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = (𝑀 − 𝑏)} ⊆ ((1...𝑁) ↑m (0..^𝑆))
25	23, 24	eqsstrdi 3980	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ⊆ ((1...𝑁) ↑m (0..^𝑆)))
26	9	elexd 3466	. . . 4 ⊢ (𝜑 → (1...𝑁) ∈ V)
27		fzonel 13603	. . . . 5 ⊢ ¬ 𝑆 ∈ (0..^𝑆)
28	27	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 𝑆 ∈ (0..^𝑆))
29	25, 26, 5, 28, 6	actfunsnrndisj 34789	. . 3 ⊢ (𝜑 → Disj 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
30		fzofi 13911	. . . . . 6 ⊢ (0..^(𝑆 + 1)) ∈ Fin
31	30	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → (0..^(𝑆 + 1)) ∈ Fin)
32		breprexplema.l	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (0..^(𝑆 + 1))) ∧ 𝑦 ∈ ℕ) → ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
33	32	ralrimiva 3130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(𝑆 + 1))) → ∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
34	33	ralrimiva 3130	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
35	34	ad3antrrr 731	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
36		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → 𝑎 ∈ (0..^(𝑆 + 1)))
37		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣(𝜑 ∧ 𝑏 ∈ (1...𝑁))
38		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣𝑑
39		nfmpt1 5199	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
40	39	nfrn 5911	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
41	38, 40	nfel 2914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
42	37, 41	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑣((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
43	1	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → (1...𝑁) ⊆ ℕ)
44	15	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → (𝑀 − 𝑏) ∈ ℤ)
45	16	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑆 ∈ ℕ0)
46		simplr 769	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)))
47	43, 44, 45, 46	reprf 34796	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑣:(0..^𝑆)⟶(1...𝑁))
48	13	ad3antrrr 731	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑏 ∈ (1...𝑁))
49	45, 48	fsnd 6828	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶(1...𝑁))
50		fzodisjsn 13627	. . . . . . . . . . . . . 14 ⊢ ((0..^𝑆) ∩ {𝑆}) = ∅
51	50	a1i 11	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → ((0..^𝑆) ∩ {𝑆}) = ∅)
52		fun2 6707	. . . . . . . . . . . . 13 ⊢ (((𝑣:(0..^𝑆)⟶(1...𝑁) ∧ {⟨𝑆, 𝑏⟩}:{𝑆}⟶(1...𝑁)) ∧ ((0..^𝑆) ∩ {𝑆}) = ∅) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}):((0..^𝑆) ∪ {𝑆})⟶(1...𝑁))
53	47, 49, 51, 52	syl21anc 838	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}):((0..^𝑆) ∪ {𝑆})⟶(1...𝑁))
54		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
55		nn0uz 12803	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 = (ℤ≥‘0)
56	5, 55	eleqtrdi 2847	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (ℤ≥‘0))
57		fzosplitsn 13706	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (ℤ≥‘0) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
58	56, 57	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
59	58	ad4antr 733	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
60	54, 59	feq12d 6660	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → (𝑑:(0..^(𝑆 + 1))⟶(1...𝑁) ↔ (𝑣 ∪ {⟨𝑆, 𝑏⟩}):((0..^𝑆) ∪ {𝑆})⟶(1...𝑁)))
61	53, 60	mpbird 257	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩})) → 𝑑:(0..^(𝑆 + 1))⟶(1...𝑁))
62		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
63		vex 3446	. . . . . . . . . . . . . 14 ⊢ 𝑣 ∈ V
64		snex 5387	. . . . . . . . . . . . . 14 ⊢ {⟨𝑆, 𝑏⟩} ∈ V
65	63, 64	unex 7701	. . . . . . . . . . . . 13 ⊢ (𝑣 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
66	6, 65	elrnmpti 5921	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ↔ ∃𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
67	62, 66	sylib 218	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → ∃𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
68	42, 61, 67	r19.29af 3247	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → 𝑑:(0..^(𝑆 + 1))⟶(1...𝑁))
69	68	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → 𝑑:(0..^(𝑆 + 1))⟶(1...𝑁))
70	69, 36	ffvelcdmd 7041	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑‘𝑎) ∈ (1...𝑁))
71	1, 70	sselid 3933	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑‘𝑎) ∈ ℕ)
72		fveq2 6844	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐿‘𝑥) = (𝐿‘𝑎))
73	72	fveq1d 6846	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝐿‘𝑥)‘𝑦) = ((𝐿‘𝑎)‘𝑦))
74	73	eleq1d 2822	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (((𝐿‘𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘𝑦) ∈ ℂ))
75		fveq2 6844	. . . . . . . . 9 ⊢ (𝑦 = (𝑑‘𝑎) → ((𝐿‘𝑎)‘𝑦) = ((𝐿‘𝑎)‘(𝑑‘𝑎)))
76	75	eleq1d 2822	. . . . . . . 8 ⊢ (𝑦 = (𝑑‘𝑎) → (((𝐿‘𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ))
77	74, 76	rspc2v 3589	. . . . . . 7 ⊢ ((𝑎 ∈ (0..^(𝑆 + 1)) ∧ (𝑑‘𝑎) ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ → ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ))
78	36, 71, 77	syl2anc 585	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ → ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ))
79	35, 78	mpd 15	. . . . 5 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
80	31, 79	fprodcl 15889	. . . 4 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
81	80	anasss 466	. . 3 ⊢ ((𝜑 ∧ (𝑏 ∈ (1...𝑁) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ)
82	9, 22, 29, 81	fsumiun 15758	. 2 ⊢ (𝜑 → Σ𝑑 ∈ ∪ 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)))
83	58	ad2antrr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
84	83	prodeq1d 15857	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = ∏𝑎 ∈ ((0..^𝑆) ∪ {𝑆})((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
85		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑎((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)))
86		nfcv 2899	. . . . . . 7 ⊢ Ⅎ𝑎((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
87		fzofi 13911	. . . . . . . 8 ⊢ (0..^𝑆) ∈ Fin
88	87	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (0..^𝑆) ∈ Fin)
89	16	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑆 ∈ ℕ0)
90	27	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ¬ 𝑆 ∈ (0..^𝑆))
91	1	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (1...𝑁) ⊆ ℕ)
92	15	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (𝑀 − 𝑏) ∈ ℤ)
93		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)))
94	91, 92, 89, 93	reprf 34796	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑒:(0..^𝑆)⟶(1...𝑁))
95	94	ffnd 6673	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑒 Fn (0..^𝑆))
96	95	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒 Fn (0..^𝑆))
97	13	adantr 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑏 ∈ (1...𝑁))
98		fnsng 6554	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑁)) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
99	89, 97, 98	syl2anc 585	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
100	99	adantr 480	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
101	50	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((0..^𝑆) ∩ {𝑆}) = ∅)
102		simpr 484	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
103		fvun1 6935	. . . . . . . . . 10 ⊢ ((𝑒 Fn (0..^𝑆) ∧ {⟨𝑆, 𝑏⟩} Fn {𝑆} ∧ (((0..^𝑆) ∩ {𝑆}) = ∅ ∧ 𝑎 ∈ (0..^𝑆))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑒‘𝑎))
104	96, 100, 101, 102, 103	syl112anc 1377	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑒‘𝑎))
105	104	fveq2d 6848	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
106	34	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
107	106	adantr 480	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
108		fzossfzop1 13673	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ ℕ0 → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
109	5, 108	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
110	109	ad2antrr 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
111	110	sselda 3935	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
112	94	ffvelcdmda 7040	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ (1...𝑁))
113	1, 112	sselid 3933	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ ℕ)
114		fveq2 6844	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑒‘𝑎) → ((𝐿‘𝑎)‘𝑦) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
115	114	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑒‘𝑎) → (((𝐿‘𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘(𝑒‘𝑎)) ∈ ℂ))
116	74, 115	rspc2v 3589	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (0..^(𝑆 + 1)) ∧ (𝑒‘𝑎) ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ → ((𝐿‘𝑎)‘(𝑒‘𝑎)) ∈ ℂ))
117	111, 113, 116	syl2anc 585	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ → ((𝐿‘𝑎)‘(𝑒‘𝑎)) ∈ ℂ))
118	107, 117	mpd 15	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑒‘𝑎)) ∈ ℂ)
119	105, 118	eqeltrd 2837	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) ∈ ℂ)
120		fveq2 6844	. . . . . . . 8 ⊢ (𝑎 = 𝑆 → (𝐿‘𝑎) = (𝐿‘𝑆))
121		fveq2 6844	. . . . . . . 8 ⊢ (𝑎 = 𝑆 → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
122	120, 121	fveq12d 6851	. . . . . . 7 ⊢ (𝑎 = 𝑆 → ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)))
123	50	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((0..^𝑆) ∩ {𝑆}) = ∅)
124		snidg 4619	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ℕ0 → 𝑆 ∈ {𝑆})
125	89, 124	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑆 ∈ {𝑆})
126		fvun2 6936	. . . . . . . . . . 11 ⊢ ((𝑒 Fn (0..^𝑆) ∧ {⟨𝑆, 𝑏⟩} Fn {𝑆} ∧ (((0..^𝑆) ∩ {𝑆}) = ∅ ∧ 𝑆 ∈ {𝑆})) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
127	95, 99, 123, 125, 126	syl112anc 1377	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
128		fvsng 7138	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑁)) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
129	89, 97, 128	syl2anc 585	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
130	127, 129	eqtrd 2772	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = 𝑏)
131	130	fveq2d 6848	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) = ((𝐿‘𝑆)‘𝑏))
132		fzonn0p1 13672	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ℕ0 → 𝑆 ∈ (0..^(𝑆 + 1)))
133	5, 132	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ (0..^(𝑆 + 1)))
134	133	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑆 ∈ (0..^(𝑆 + 1)))
135	1, 97	sselid 3933	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑏 ∈ ℕ)
136		fveq2 6844	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑆 → (𝐿‘𝑥) = (𝐿‘𝑆))
137	136	fveq1d 6846	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑆 → ((𝐿‘𝑥)‘𝑦) = ((𝐿‘𝑆)‘𝑦))
138	137	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑆 → (((𝐿‘𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑆)‘𝑦) ∈ ℂ))
139		fveq2 6844	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → ((𝐿‘𝑆)‘𝑦) = ((𝐿‘𝑆)‘𝑏))
140	139	eleq1d 2822	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (((𝐿‘𝑆)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑆)‘𝑏) ∈ ℂ))
141	138, 140	rspc2v 3589	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (0..^(𝑆 + 1)) ∧ 𝑏 ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ → ((𝐿‘𝑆)‘𝑏) ∈ ℂ))
142	134, 135, 141	syl2anc 585	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ → ((𝐿‘𝑆)‘𝑏) ∈ ℂ))
143	106, 142	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝐿‘𝑆)‘𝑏) ∈ ℂ)
144	131, 143	eqeltrd 2837	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) ∈ ℂ)
145	85, 86, 88, 89, 90, 119, 122, 144	fprodsplitsn 15926	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ∏𝑎 ∈ ((0..^𝑆) ∪ {𝑆})((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) · ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))))
146	105	prodeq2dv 15859	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)))
147	146, 131	oveq12d 7388	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) · ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)) · ((𝐿‘𝑆)‘𝑏)))
148	84, 145, 147	3eqtrd 2776	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)) · ((𝐿‘𝑆)‘𝑏)))
149	148	sumeq2dv 15639	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → Σ𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = Σ𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)) · ((𝐿‘𝑆)‘𝑏)))
150		simpl 482	. . . . . . . 8 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → 𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
151	150	fveq1d 6846	. . . . . . 7 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎))
152	151	fveq2d 6848	. . . . . 6 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) = ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
153	152	prodeq2dv 15859	. . . . 5 ⊢ (𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
154	25, 26, 5, 28, 6	actfunsnf1o 34788	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})):((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))–1-1-onto→ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
155	6	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) = (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
156		simpr 484	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑣 = 𝑒) → 𝑣 = 𝑒)
157	156	uneq1d 4121	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑣 = 𝑒) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}) = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
158		vex 3446	. . . . . . . 8 ⊢ 𝑒 ∈ V
159	158, 64	unex 7701	. . . . . . 7 ⊢ (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
160	159	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∈ V)
161	155, 157, 93, 160	fvmptd 6959	. . . . 5 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))‘𝑒) = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
162	153, 18, 154, 161, 80	fsumf1o 15660	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → Σ𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
163		simpl 482	. . . . . . . . . 10 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → 𝑑 = 𝑒)
164	163	fveq1d 6846	. . . . . . . . 9 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → (𝑑‘𝑎) = (𝑒‘𝑎))
165	164	fveq2d 6848	. . . . . . . 8 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
166	165	prodeq2dv 15859	. . . . . . 7 ⊢ (𝑑 = 𝑒 → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)))
167	166	oveq1d 7385	. . . . . 6 ⊢ (𝑑 = 𝑒 → (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑆)‘𝑏)) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)) · ((𝐿‘𝑆)‘𝑏)))
168	167	cbvsumv 15633	. . . . 5 ⊢ Σ𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑆)‘𝑏)) = Σ𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)) · ((𝐿‘𝑆)‘𝑏))
169	168	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → Σ𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑆)‘𝑏)) = Σ𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)) · ((𝐿‘𝑆)‘𝑏)))
170	149, 162, 169	3eqtr4d 2782	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → Σ𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑆)‘𝑏)))
171	170	sumeq2dv 15639	. 2 ⊢ (𝜑 → Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑆)‘𝑏)))
172	8, 82, 171	3eqtrd 2776	1 ⊢ (𝜑 → Σ𝑑 ∈ ((1...𝑁)(repr‘(𝑆 + 1))𝑀)∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = Σ𝑏 ∈ (1...𝑁)Σ𝑑 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))(∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑆)‘𝑏)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  {crab 3401  Vcvv 3442   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582  ⟨cop 4588  ∪ ciun 4948   class class class wbr 5100   ↦ cmpt 5181  ran crn 5635   Fn wfn 6497  ⟶wf 6498  ‘cfv 6502  (class class class)co 7370   ↑_m cmap 8777  Fincfn 8897  ℂcc 11038  0cc0 11040  1c1 11041   + caddc 11043   · cmul 11045   ≤ cle 11181   − cmin 11378  ℕcn 12159  ℕ₀cn0 12415  ℤcz 12502  ℤ_≥cuz 12765  ...cfz 13437  ..^cfzo 13584  Σcsu 15623  ∏cprod 15840  reprcrepr 34792

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692  ax-inf2 9564  ax-cnex 11096  ax-resscn 11097  ax-1cn 11098  ax-icn 11099  ax-addcl 11100  ax-addrcl 11101  ax-mulcl 11102  ax-mulrcl 11103  ax-mulcom 11104  ax-addass 11105  ax-mulass 11106  ax-distr 11107  ax-i2m1 11108  ax-1ne0 11109  ax-1rid 11110  ax-rnegex 11111  ax-rrecex 11112  ax-cnre 11113  ax-pre-lttri 11114  ax-pre-lttrn 11115  ax-pre-ltadd 11116  ax-pre-mulgt0 11117  ax-pre-sup 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-disj 5068  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-se 5588  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6269  df-ord 6330  df-on 6331  df-lim 6332  df-suc 6333  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-isom 6511  df-riota 7327  df-ov 7373  df-oprab 7374  df-mpo 7375  df-om 7821  df-1st 7945  df-2nd 7946  df-frecs 8235  df-wrecs 8266  df-recs 8315  df-rdg 8353  df-1o 8409  df-er 8647  df-map 8779  df-pm 8780  df-en 8898  df-dom 8899  df-sdom 8900  df-fin 8901  df-sup 9359  df-oi 9429  df-card 9865  df-pnf 11182  df-mnf 11183  df-xr 11184  df-ltxr 11185  df-le 11186  df-sub 11380  df-neg 11381  df-div 11809  df-nn 12160  df-2 12222  df-3 12223  df-n0 12416  df-z 12503  df-uz 12766  df-rp 12920  df-fz 13438  df-fzo 13585  df-seq 13939  df-exp 13999  df-hash 14268  df-cj 15036  df-re 15037  df-im 15038  df-sqrt 15172  df-abs 15173  df-clim 15425  df-sum 15624  df-prod 15841  df-repr 34793

This theorem is referenced by:  breprexplemc  34816