Theorem breprexplema 32619

Description: Lemma for breprexp 32622 (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 13296	. . . . 5 ⊢ (1...𝑁) ⊆ ℕ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
3		breprexplema.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
4	3	nn0zd 12433	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		breprexp.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
6		eqid 2739	. . . 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 32604	. . 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 15422	. 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 13702	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
10	1	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (1...𝑁) ⊆ ℕ)
11	4	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑀 ∈ ℤ)
12		fzssz 13267	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℤ
13		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
14	12, 13	sselid 3920	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℤ)
15	11, 14	zsubcld 12440	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (𝑀 − 𝑏) ∈ ℤ)
16	5	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑆 ∈ ℕ0)
17	9	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (1...𝑁) ∈ Fin)
18	10, 15, 16, 17	reprfi 32605	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ∈ Fin)
19		mptfi 9127	. . . . 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 9111	. . . 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 32599	. . . . 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 4014	. . . . 5 ⊢ {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = (𝑀 − 𝑏)} ⊆ ((1...𝑁) ↑m (0..^𝑆))
25	23, 24	eqsstrdi 3976	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ⊆ ((1...𝑁) ↑m (0..^𝑆)))
26	9	elexd 3453	. . . 4 ⊢ (𝜑 → (1...𝑁) ∈ V)
27		fzonel 13410	. . . . 5 ⊢ ¬ 𝑆 ∈ (0..^𝑆)
28	27	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 𝑆 ∈ (0..^𝑆))
29	25, 26, 5, 28, 6	actfunsnrndisj 32594	. . 3 ⊢ (𝜑 → Disj 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
30		fzofi 13703	. . . . . 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 3104	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(𝑆 + 1))) → ∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
34	33	ralrimiva 3104	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
35	34	ad3antrrr 727	. . . . . 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 485	. . . . . . 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 1918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣(𝜑 ∧ 𝑏 ∈ (1...𝑁))
38		nfcv 2908	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣𝑑
39		nfmpt1 5183	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
40	39	nfrn 5864	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
41	38, 40	nfel 2922	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
42	37, 41	nfan 1903	. . . . . . . . . . 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 727	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . 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 32601	. . . . . . . . . . . . 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 727	. . . . . . . . . . . . . 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 6768	. . . . . . . . . . . . 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 13434	. . . . . . . . . . . . . 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 6646	. . . . . . . . . . . . 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 835	. . . . . . . . . . . 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 485	. . . . . . . . . . . . 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 12629	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 = (ℤ≥‘0)
56	5, 55	eleqtrdi 2850	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (ℤ≥‘0))
57		fzosplitsn 13504	. . . . . . . . . . . . . . 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 729	. . . . . . . . . . . . 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 6597	. . . . . . . . . . . 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 256	. . . . . . . . . . 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 485	. . . . . . . . . . . 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 3437	. . . . . . . . . . . . . 14 ⊢ 𝑣 ∈ V
64		snex 5355	. . . . . . . . . . . . . 14 ⊢ {⟨𝑆, 𝑏⟩} ∈ V
65	63, 64	unex 7605	. . . . . . . . . . . . 13 ⊢ (𝑣 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
66	6, 65	elrnmpti 5872	. . . . . . . . . . . 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 217	. . . . . . . . . . 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 3263	. . . . . . . . . 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 481	. . . . . . . . 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	ffvelrnd 6971	. . . . . . . 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 3920	. . . . . . 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 6783	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐿‘𝑥) = (𝐿‘𝑎))
73	72	fveq1d 6785	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝐿‘𝑥)‘𝑦) = ((𝐿‘𝑎)‘𝑦))
74	73	eleq1d 2824	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (((𝐿‘𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘𝑦) ∈ ℂ))
75		fveq2 6783	. . . . . . . . 9 ⊢ (𝑦 = (𝑑‘𝑎) → ((𝐿‘𝑎)‘𝑦) = ((𝐿‘𝑎)‘(𝑑‘𝑎)))
76	75	eleq1d 2824	. . . . . . . 8 ⊢ (𝑦 = (𝑑‘𝑎) → (((𝐿‘𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ))
77	74, 76	rspc2v 3571	. . . . . . 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 584	. . . . . 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 15671	. . . 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 467	. . 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 15542	. 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 723	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
84	83	prodeq1d 15640	. . . . . 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 1918	. . . . . . 7 ⊢ Ⅎ𝑎((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)))
86		nfcv 2908	. . . . . . 7 ⊢ Ⅎ𝑎((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
87		fzofi 13703	. . . . . . . 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 481	. . . . . . 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 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (𝑀 − 𝑏) ∈ ℤ)
93		simpr 485	. . . . . . . . . . . . 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 32601	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑒:(0..^𝑆)⟶(1...𝑁))
95	94	ffnd 6610	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑒 Fn (0..^𝑆))
96	95	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒 Fn (0..^𝑆))
97	13	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑏 ∈ (1...𝑁))
98		fnsng 6493	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑁)) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
99	89, 97, 98	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
100	99	adantr 481	. . . . . . . . . 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 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
103		fvun1 6868	. . . . . . . . . 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 1373	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑒‘𝑎))
105	104	fveq2d 6787	. . . . . . . 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 723	. . . . . . . . . 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 481	. . . . . . . . 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 13474	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ ℕ0 → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
109	5, 108	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
110	109	ad2antrr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
111	110	sselda 3922	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
112	94	ffvelrnda 6970	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ (1...𝑁))
113	1, 112	sselid 3920	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ ℕ)
114		fveq2 6783	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑒‘𝑎) → ((𝐿‘𝑎)‘𝑦) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
115	114	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑒‘𝑎) → (((𝐿‘𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘(𝑒‘𝑎)) ∈ ℂ))
116	74, 115	rspc2v 3571	. . . . . . . . . 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 584	. . . . . . . . 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 2840	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) ∈ ℂ)
120		fveq2 6783	. . . . . . . 8 ⊢ (𝑎 = 𝑆 → (𝐿‘𝑎) = (𝐿‘𝑆))
121		fveq2 6783	. . . . . . . 8 ⊢ (𝑎 = 𝑆 → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
122	120, 121	fveq12d 6790	. . . . . . 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 4596	. . . . . . . . . . . 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 6869	. . . . . . . . . . 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 1373	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
128		fvsng 7061	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑁)) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
129	89, 97, 128	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
130	127, 129	eqtrd 2779	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = 𝑏)
131	130	fveq2d 6787	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) = ((𝐿‘𝑆)‘𝑏))
132		fzonn0p1 13473	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ℕ0 → 𝑆 ∈ (0..^(𝑆 + 1)))
133	5, 132	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ (0..^(𝑆 + 1)))
134	133	ad2antrr 723	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑆 ∈ (0..^(𝑆 + 1)))
135	1, 97	sselid 3920	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑏 ∈ ℕ)
136		fveq2 6783	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑆 → (𝐿‘𝑥) = (𝐿‘𝑆))
137	136	fveq1d 6785	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑆 → ((𝐿‘𝑥)‘𝑦) = ((𝐿‘𝑆)‘𝑦))
138	137	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑆 → (((𝐿‘𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑆)‘𝑦) ∈ ℂ))
139		fveq2 6783	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → ((𝐿‘𝑆)‘𝑦) = ((𝐿‘𝑆)‘𝑏))
140	139	eleq1d 2824	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (((𝐿‘𝑆)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑆)‘𝑏) ∈ ℂ))
141	138, 140	rspc2v 3571	. . . . . . . . . 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 584	. . . . . . . . 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 2840	. . . . . . 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 15708	. . . . . 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 15642	. . . . . . 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 7302	. . . . . 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 2783	. . . . 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 15424	. . . 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 483	. . . . . . . 8 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → 𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
151	150	fveq1d 6785	. . . . . . 7 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎))
152	151	fveq2d 6787	. . . . . 6 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) = ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
153	152	prodeq2dv 15642	. . . . 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 32593	. . . . 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 485	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑣 = 𝑒) → 𝑣 = 𝑒)
157	156	uneq1d 4097	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑣 = 𝑒) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}) = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
158		vex 3437	. . . . . . . 8 ⊢ 𝑒 ∈ V
159	158, 64	unex 7605	. . . . . . 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 6891	. . . . 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 15444	. . . 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 483	. . . . . . . . . 10 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → 𝑑 = 𝑒)
164	163	fveq1d 6785	. . . . . . . . 9 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → (𝑑‘𝑎) = (𝑒‘𝑎))
165	164	fveq2d 6787	. . . . . . . 8 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
166	165	prodeq2dv 15642	. . . . . . 7 ⊢ (𝑑 = 𝑒 → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)))
167	166	oveq1d 7299	. . . . . 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 15417	. . . . 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 2789	. . 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 15424	. 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 2783	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 396   = wceq 1539   ∈ wcel 2107  ∀wral 3065  ∃wrex 3066  {crab 3069  Vcvv 3433   ∪ cun 3886   ∩ cin 3887   ⊆ wss 3888  ∅c0 4257  {csn 4562  ⟨cop 4568  ∪ ciun 4925   class class class wbr 5075   ↦ cmpt 5158  ran crn 5591   Fn wfn 6432  ⟶wf 6433  ‘cfv 6437  (class class class)co 7284   ↑_m cmap 8624  Fincfn 8742  ℂcc 10878  0cc0 10880  1c1 10881   + caddc 10883   · cmul 10885   ≤ cle 11019   − cmin 11214  ℕcn 11982  ℕ₀cn0 12242  ℤcz 12328  ℤ_≥cuz 12591  ...cfz 13248  ..^cfzo 13391  Σcsu 15406  ∏cprod 15624  reprcrepr 32597

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597  ax-inf2 9408  ax-cnex 10936  ax-resscn 10937  ax-1cn 10938  ax-icn 10939  ax-addcl 10940  ax-addrcl 10941  ax-mulcl 10942  ax-mulrcl 10943  ax-mulcom 10944  ax-addass 10945  ax-mulass 10946  ax-distr 10947  ax-i2m1 10948  ax-1ne0 10949  ax-1rid 10950  ax-rnegex 10951  ax-rrecex 10952  ax-cnre 10953  ax-pre-lttri 10954  ax-pre-lttrn 10955  ax-pre-ltadd 10956  ax-pre-mulgt0 10957  ax-pre-sup 10958

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-disj 5041  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-se 5546  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6206  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-isom 6446  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-frecs 8106  df-wrecs 8137  df-recs 8211  df-rdg 8250  df-1o 8306  df-er 8507  df-map 8626  df-pm 8627  df-en 8743  df-dom 8744  df-sdom 8745  df-fin 8746  df-sup 9210  df-oi 9278  df-card 9706  df-pnf 11020  df-mnf 11021  df-xr 11022  df-ltxr 11023  df-le 11024  df-sub 11216  df-neg 11217  df-div 11642  df-nn 11983  df-2 12045  df-3 12046  df-n0 12243  df-z 12329  df-uz 12592  df-rp 12740  df-fz 13249  df-fzo 13392  df-seq 13731  df-exp 13792  df-hash 14054  df-cj 14819  df-re 14820  df-im 14821  df-sqrt 14955  df-abs 14956  df-clim 15206  df-sum 15407  df-prod 15625  df-repr 32598

This theorem is referenced by:  breprexplemc  32621