Theorem breprexplema 34964

Description: Lemma for breprexp 34967 (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 13585	. . . . 5 ⊢ (1...𝑁) ⊆ ℕ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
3		breprexplema.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
4	3	nn0zd 12618	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		breprexp.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
6		eqid 2769	. . . 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 34949	. . 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 15753	. 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 14011	. . 3 ⊢ (𝜑 → (1...𝑁) ∈ Fin)
10	1	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (1...𝑁) ⊆ ℕ)
11	4	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑀 ∈ ℤ)
12		fzssz 13556	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℤ
13		simpr 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
14	12, 13	sselid 3943	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℤ)
15	11, 14	zsubcld 12707	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (𝑀 − 𝑏) ∈ ℤ)
16	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑆 ∈ ℕ0)
17	9	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (1...𝑁) ∈ Fin)
18	10, 15, 16, 17	reprfi 34950	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ∈ Fin)
19		mptfi 9310	. . . . 5 ⊢ (((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ∈ Fin → (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
20	18, 19	syl 18	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
21		rnfi 9299	. . . 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 18	. . 3 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ∈ Fin)
23	10, 15, 16	reprval 34944	. . . . 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 4042	. . . . 5 ⊢ {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = (𝑀 − 𝑏)} ⊆ ((1...𝑁) ↑m (0..^𝑆))
25	23, 24	eqsstrdi 3989	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ⊆ ((1...𝑁) ↑m (0..^𝑆)))
26	9	elexd 3486	. . . 4 ⊢ (𝜑 → (1...𝑁) ∈ V)
27		fzonel 13704	. . . . 5 ⊢ ¬ 𝑆 ∈ (0..^𝑆)
28	27	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 𝑆 ∈ (0..^𝑆))
29	25, 26, 5, 28, 6	actfunsnrndisj 34939	. . 3 ⊢ (𝜑 → Disj 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
30		fzofi 14012	. . . . . 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 3163	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(𝑆 + 1))) → ∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
34	33	ralrimiva 3163	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
35	34	ad3antrrr 742	. . . . . 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 489	. . . . . . 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 1941	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣(𝜑 ∧ 𝑏 ∈ (1...𝑁))
38		nfcv 2931	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣𝑑
39		nfmpt1 5214	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
40	39	nfrn 5945	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
41	38, 40	nfel 2945	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
42	37, 41	nfan 1926	. . . . . . . . . . 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 742	. . . . . . . . . . . . . 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 742	. . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . 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 34946	. . . . . . . . . . . . 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 742	. . . . . . . . . . . . . 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 6868	. . . . . . . . . . . . 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 13728	. . . . . . . . . . . . . 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 6744	. . . . . . . . . . . . 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 850	. . . . . . . . . . . 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 489	. . . . . . . . . . . . 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 12902	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 = (ℤ≥‘0)
56	5, 55	eleqtrdi 2879	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (ℤ≥‘0))
57		fzosplitsn 13807	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ (ℤ≥‘0) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
58	56, 57	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
59	58	ad4antr 744	. . . . . . . . . . . . 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 6696	. . . . . . . . . . . 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 260	. . . . . . . . . . 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		vex 3467	. . . . . . . . . . . . . 14 ⊢ 𝑣 ∈ V
63		snex 5413	. . . . . . . . . . . . . 14 ⊢ {⟨𝑆, 𝑏⟩} ∈ V
64	62, 63	unex 7745	. . . . . . . . . . . . 13 ⊢ (𝑣 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
65	6, 64	elrnmpti 5955	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})) ↔ ∃𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))𝑑 = (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
66	65	bilani 509	. . . . . . . . . . 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⟩}))
67	42, 61, 66	r19.29af 3280	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) → 𝑑:(0..^(𝑆 + 1))⟶(1...𝑁))
68	67	adantr 485	. . . . . . . . 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))
69	68, 36	ffvelcdmd 7083	. . . . . . . 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))
70	1, 69	sselid 3943	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑‘𝑎) ∈ ℕ)
71		fveq2 6884	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐿‘𝑥) = (𝐿‘𝑎))
72	71	fveq1d 6886	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝐿‘𝑥)‘𝑦) = ((𝐿‘𝑎)‘𝑦))
73	72	eleq1d 2854	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (((𝐿‘𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘𝑦) ∈ ℂ))
74		fveq2 6884	. . . . . . . . 9 ⊢ (𝑦 = (𝑑‘𝑎) → ((𝐿‘𝑎)‘𝑦) = ((𝐿‘𝑎)‘(𝑑‘𝑎)))
75	74	eleq1d 2854	. . . . . . . 8 ⊢ (𝑦 = (𝑑‘𝑎) → (((𝐿‘𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ))
76	73, 75	rspc2v 3601	. . . . . . 7 ⊢ ((𝑎 ∈ (0..^(𝑆 + 1)) ∧ (𝑑‘𝑎) ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ → ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ))
77	36, 70, 76	syl2anc 595	. . . . . 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))
78	35, 77	mpd 16	. . . . 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)
79	31, 78	fprodcl 16008	. . . 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)
80	79	anasss 471	. . 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)
81	9, 22, 29, 80	fsumiun 15875	. 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)))
82	58	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
83	82	prodeq1d 15976	. . . . . 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)))
84		nfv 1941	. . . . . . 7 ⊢ Ⅎ𝑎((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)))
85		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑎((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
86		fzofi 14012	. . . . . . . 8 ⊢ (0..^𝑆) ∈ Fin
87	86	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (0..^𝑆) ∈ Fin)
88	16	adantr 485	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑆 ∈ ℕ0)
89	27	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ¬ 𝑆 ∈ (0..^𝑆))
90	1	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (1...𝑁) ⊆ ℕ)
91	15	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (𝑀 − 𝑏) ∈ ℤ)
92		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)))
93	90, 91, 88, 92	reprf 34946	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑒:(0..^𝑆)⟶(1...𝑁))
94	93	ffnd 6709	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑒 Fn (0..^𝑆))
95	94	adantr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒 Fn (0..^𝑆))
96	13	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑏 ∈ (1...𝑁))
97		fnsng 6591	. . . . . . . . . . . 12 ⊢ ((𝑆 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑁)) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
98	88, 96, 97	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
99	98	adantr 485	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
100	50	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((0..^𝑆) ∩ {𝑆}) = ∅)
101		simpr 489	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
102		fvun1 6975	. . . . . . . . . 10 ⊢ ((𝑒 Fn (0..^𝑆) ∧ {⟨𝑆, 𝑏⟩} Fn {𝑆} ∧ (((0..^𝑆) ∩ {𝑆}) = ∅ ∧ 𝑎 ∈ (0..^𝑆))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑒‘𝑎))
103	95, 99, 100, 101, 102	syl112anc 1399	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑒‘𝑎))
104	103	fveq2d 6888	. . . . . . . 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)))
105	34	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
106	105	adantr 485	. . . . . . . . 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)
107		fzossfzop1 13774	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ ℕ0 → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
108	5, 107	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
109	108	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
110	109	sselda 3945	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
111	93	ffvelcdmda 7082	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ (1...𝑁))
112	1, 111	sselid 3943	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ ℕ)
113		fveq2 6884	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑒‘𝑎) → ((𝐿‘𝑎)‘𝑦) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
114	113	eleq1d 2854	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑒‘𝑎) → (((𝐿‘𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘(𝑒‘𝑎)) ∈ ℂ))
115	73, 114	rspc2v 3601	. . . . . . . . . 10 ⊢ ((𝑎 ∈ (0..^(𝑆 + 1)) ∧ (𝑒‘𝑎) ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ → ((𝐿‘𝑎)‘(𝑒‘𝑎)) ∈ ℂ))
116	110, 112, 115	syl2anc 595	. . . . . . . . 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))
117	106, 116	mpd 16	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑒‘𝑎)) ∈ ℂ)
118	104, 117	eqeltrd 2869	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) ∈ ℂ)
119		fveq2 6884	. . . . . . . 8 ⊢ (𝑎 = 𝑆 → (𝐿‘𝑎) = (𝐿‘𝑆))
120		fveq2 6884	. . . . . . . 8 ⊢ (𝑎 = 𝑆 → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
121	119, 120	fveq12d 6891	. . . . . . 7 ⊢ (𝑎 = 𝑆 → ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) = ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)))
122	50	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((0..^𝑆) ∩ {𝑆}) = ∅)
123		snidg 4631	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ℕ0 → 𝑆 ∈ {𝑆})
124	88, 123	syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑆 ∈ {𝑆})
125		fvun2 6976	. . . . . . . . . . 11 ⊢ ((𝑒 Fn (0..^𝑆) ∧ {⟨𝑆, 𝑏⟩} Fn {𝑆} ∧ (((0..^𝑆) ∩ {𝑆}) = ∅ ∧ 𝑆 ∈ {𝑆})) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
126	94, 98, 122, 124, 125	syl112anc 1399	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
127		fvsng 7181	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ ℕ0 ∧ 𝑏 ∈ (1...𝑁)) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
128	88, 96, 127	syl2anc 595	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
129	126, 128	eqtrd 2804	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = 𝑏)
130	129	fveq2d 6888	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) = ((𝐿‘𝑆)‘𝑏))
131		fzonn0p1 13773	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ℕ0 → 𝑆 ∈ (0..^(𝑆 + 1)))
132	5, 131	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ (0..^(𝑆 + 1)))
133	132	ad2antrr 738	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑆 ∈ (0..^(𝑆 + 1)))
134	1, 96	sselid 3943	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑏 ∈ ℕ)
135		fveq2 6884	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑆 → (𝐿‘𝑥) = (𝐿‘𝑆))
136	135	fveq1d 6886	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑆 → ((𝐿‘𝑥)‘𝑦) = ((𝐿‘𝑆)‘𝑦))
137	136	eleq1d 2854	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑆 → (((𝐿‘𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑆)‘𝑦) ∈ ℂ))
138		fveq2 6884	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → ((𝐿‘𝑆)‘𝑦) = ((𝐿‘𝑆)‘𝑏))
139	138	eleq1d 2854	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (((𝐿‘𝑆)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑆)‘𝑏) ∈ ℂ))
140	137, 139	rspc2v 3601	. . . . . . . . . 10 ⊢ ((𝑆 ∈ (0..^(𝑆 + 1)) ∧ 𝑏 ∈ ℕ) → (∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ → ((𝐿‘𝑆)‘𝑏) ∈ ℂ))
141	133, 134, 140	syl2anc 595	. . . . . . . . 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))
142	105, 141	mpd 16	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝐿‘𝑆)‘𝑏) ∈ ℂ)
143	130, 142	eqeltrd 2869	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) ∈ ℂ)
144	84, 85, 87, 88, 89, 118, 121, 143	fprodsplitsn 16045	. . . . . 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))))
145	104	prodeq2dv 15978	. . . . . . 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)))
146	145, 130	oveq12d 7431	. . . . . 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)))
147	83, 144, 146	3eqtrd 2808	. . . . 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)))
148	147	sumeq2dv 15755	. . . 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)))
149		simpl 487	. . . . . . . 8 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → 𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
150	149	fveq1d 6886	. . . . . . 7 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎))
151	150	fveq2d 6888	. . . . . 6 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) = ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
152	151	prodeq2dv 15978	. . . . 5 ⊢ (𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) → ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘(𝑑‘𝑎)) = ∏𝑎 ∈ (0..^(𝑆 + 1))((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
153	25, 26, 5, 28, 6	actfunsnf1o 34938	. . . . 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⟩})))
154	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⟩})))
155		simpr 489	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑣 = 𝑒) → 𝑣 = 𝑒)
156	155	uneq1d 4129	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑣 = 𝑒) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}) = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
157		vex 3467	. . . . . . . 8 ⊢ 𝑒 ∈ V
158	157, 63	unex 7745	. . . . . . 7 ⊢ (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
159	158	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∈ V)
160	154, 156, 92, 159	fvmptd 7000	. . . . 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⟩}))
161	152, 18, 153, 160, 79	fsumf1o 15776	. . . 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)))
162		simpl 487	. . . . . . . . . 10 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → 𝑑 = 𝑒)
163	162	fveq1d 6886	. . . . . . . . 9 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → (𝑑‘𝑎) = (𝑒‘𝑎))
164	163	fveq2d 6888	. . . . . . . 8 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
165	164	prodeq2dv 15978	. . . . . . 7 ⊢ (𝑑 = 𝑒 → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)))
166	165	oveq1d 7428	. . . . . 6 ⊢ (𝑑 = 𝑒 → (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) · ((𝐿‘𝑆)‘𝑏)) = (∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)) · ((𝐿‘𝑆)‘𝑏)))
167	166	cbvsumv 15749	. . . . 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))
168	167	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)))
169	148, 161, 168	3eqtr4d 2814	. . 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)))
170	169	sumeq2dv 15755	. 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)))
171	8, 81, 170	3eqtrd 2808	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 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3423  Vcvv 3463   ∪ cun 3911   ∩ cin 3912   ⊆ wss 3913  ∅c0 4294  {csn 4594  ⟨cop 4600  ∪ ciun 4960   class class class wbr 5113   ↦ cmpt 5196  ran crn 5665   Fn wfn 6534  ⟶wf 6535  ‘cfv 6539  (class class class)co 7413   ↑_m cmap 8826  Fincfn 8945  ℂcc 11100  0cc0 11102  1c1 11103   + caddc 11105   · cmul 11107   ≤ cle 11246   − cmin 11443  ℕcn 12235  ℕ₀cn0 12506  ℤcz 12593  ℤ_≥cuz 12864  ...cfz 13537  ..^cfzo 13684  Σcsu 15739  ∏cprod 15959  reprcrepr 34942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-inf2 9612  ax-cnex 11158  ax-resscn 11159  ax-1cn 11160  ax-icn 11161  ax-addcl 11162  ax-addrcl 11163  ax-mulcl 11164  ax-mulrcl 11165  ax-mulcom 11166  ax-addass 11167  ax-mulass 11168  ax-distr 11169  ax-i2m1 11170  ax-1ne0 11171  ax-1rid 11172  ax-rnegex 11173  ax-rrecex 11174  ax-cnre 11175  ax-pre-lttri 11176  ax-pre-lttrn 11177  ax-pre-ltadd 11178  ax-pre-mulgt0 11179  ax-pre-sup 11180

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-disj 5081  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-er 8696  df-map 8828  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-sup 9404  df-oi 9474  df-card 9927  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11445  df-neg 11446  df-div 11874  df-nn 12236  df-2 12305  df-3 12306  df-n0 12507  df-z 12594  df-uz 12865  df-rp 13019  df-fz 13538  df-fzo 13685  df-seq 14040  df-exp 14100  df-hash 14369  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-clim 15541  df-sum 15740  df-prod 15960  df-repr 34943

This theorem is referenced by:  breprexplemc  34966