Theorem breprexplema 34667

Description: Lemma for breprexp 34670 (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 13577	. . . . 5 ⊢ (1...𝑁) ⊆ ℕ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
3		breprexplema.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
4	3	nn0zd 12619	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		breprexp.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ0)
6		eqid 2736	. . . 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 34652	. . 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 15721	. 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 13996	. . 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 13548	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℤ
13		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
14	12, 13	sselid 3961	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℤ)
15	11, 14	zsubcld 12707	. . . . . 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 34653	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ∈ Fin)
19		mptfi 9368	. . . . 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 9357	. . . 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 34647	. . . . 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 4060	. . . . 5 ⊢ {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = (𝑀 − 𝑏)} ⊆ ((1...𝑁) ↑m (0..^𝑆))
25	23, 24	eqsstrdi 4008	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ⊆ ((1...𝑁) ↑m (0..^𝑆)))
26	9	elexd 3488	. . . 4 ⊢ (𝜑 → (1...𝑁) ∈ V)
27		fzonel 13695	. . . . 5 ⊢ ¬ 𝑆 ∈ (0..^𝑆)
28	27	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 𝑆 ∈ (0..^𝑆))
29	25, 26, 5, 28, 6	actfunsnrndisj 34642	. . 3 ⊢ (𝜑 → Disj 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
30		fzofi 13997	. . . . . 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 3133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(𝑆 + 1))) → ∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
34	33	ralrimiva 3133	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (0..^(𝑆 + 1))∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
35	34	ad3antrrr 730	. . . . . 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 1914	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣(𝜑 ∧ 𝑏 ∈ (1...𝑁))
38		nfcv 2899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣𝑑
39		nfmpt1 5225	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
40	39	nfrn 5937	. . . . . . . . . . . . 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 1899	. . . . . . . . . . 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 730	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . 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 34649	. . . . . . . . . . . . 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 730	. . . . . . . . . . . . . 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 6866	. . . . . . . . . . . . 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 13719	. . . . . . . . . . . . . 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 6746	. . . . . . . . . . . . 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 837	. . . . . . . . . . . 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 12899	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 = (ℤ≥‘0)
56	5, 55	eleqtrdi 2845	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (ℤ≥‘0))
57		fzosplitsn 13796	. . . . . . . . . . . . . . 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 732	. . . . . . . . . . . . 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 6699	. . . . . . . . . . . 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 3468	. . . . . . . . . . . . . 14 ⊢ 𝑣 ∈ V
64		snex 5411	. . . . . . . . . . . . . 14 ⊢ {⟨𝑆, 𝑏⟩} ∈ V
65	63, 64	unex 7743	. . . . . . . . . . . . 13 ⊢ (𝑣 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
66	6, 65	elrnmpti 5947	. . . . . . . . . . . 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 3255	. . . . . . . . . 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 7080	. . . . . . . 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 3961	. . . . . . 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 6881	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐿‘𝑥) = (𝐿‘𝑎))
73	72	fveq1d 6883	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝐿‘𝑥)‘𝑦) = ((𝐿‘𝑎)‘𝑦))
74	73	eleq1d 2820	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (((𝐿‘𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘𝑦) ∈ ℂ))
75		fveq2 6881	. . . . . . . . 9 ⊢ (𝑦 = (𝑑‘𝑎) → ((𝐿‘𝑎)‘𝑦) = ((𝐿‘𝑎)‘(𝑑‘𝑎)))
76	75	eleq1d 2820	. . . . . . . 8 ⊢ (𝑦 = (𝑑‘𝑎) → (((𝐿‘𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ))
77	74, 76	rspc2v 3617	. . . . . . 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 15973	. . . 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 15842	. 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 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
84	83	prodeq1d 15941	. . . . . 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 1914	. . . . . . 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 13997	. . . . . . . 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 34649	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑒:(0..^𝑆)⟶(1...𝑁))
95	94	ffnd 6712	. . . . . . . . . . 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 6593	. . . . . . . . . . . 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 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 6975	. . . . . . . . . 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 1376	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑒‘𝑎))
105	104	fveq2d 6885	. . . . . . . 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 726	. . . . . . . . . 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 13764	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ ℕ0 → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
109	5, 108	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
110	109	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
111	110	sselda 3963	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
112	94	ffvelcdmda 7079	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ (1...𝑁))
113	1, 112	sselid 3961	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ ℕ)
114		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑒‘𝑎) → ((𝐿‘𝑎)‘𝑦) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
115	114	eleq1d 2820	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑒‘𝑎) → (((𝐿‘𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘(𝑒‘𝑎)) ∈ ℂ))
116	74, 115	rspc2v 3617	. . . . . . . . . 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 2835	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)) ∈ ℂ)
120		fveq2 6881	. . . . . . . 8 ⊢ (𝑎 = 𝑆 → (𝐿‘𝑎) = (𝐿‘𝑆))
121		fveq2 6881	. . . . . . . 8 ⊢ (𝑎 = 𝑆 → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
122	120, 121	fveq12d 6888	. . . . . . 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 4641	. . . . . . . . . . . 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 6976	. . . . . . . . . . 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 1376	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
128		fvsng 7177	. . . . . . . . . . 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 2771	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = 𝑏)
131	130	fveq2d 6885	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) = ((𝐿‘𝑆)‘𝑏))
132		fzonn0p1 13763	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ ℕ0 → 𝑆 ∈ (0..^(𝑆 + 1)))
133	5, 132	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑆 ∈ (0..^(𝑆 + 1)))
134	133	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑆 ∈ (0..^(𝑆 + 1)))
135	1, 97	sselid 3961	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑏 ∈ ℕ)
136		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑆 → (𝐿‘𝑥) = (𝐿‘𝑆))
137	136	fveq1d 6883	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑆 → ((𝐿‘𝑥)‘𝑦) = ((𝐿‘𝑆)‘𝑦))
138	137	eleq1d 2820	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑆 → (((𝐿‘𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑆)‘𝑦) ∈ ℂ))
139		fveq2 6881	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → ((𝐿‘𝑆)‘𝑦) = ((𝐿‘𝑆)‘𝑏))
140	139	eleq1d 2820	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (((𝐿‘𝑆)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑆)‘𝑏) ∈ ℂ))
141	138, 140	rspc2v 3617	. . . . . . . . . 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 2835	. . . . . . 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 16010	. . . . . 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 15943	. . . . . . 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 7428	. . . . . 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 2775	. . . . 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 15723	. . . 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 6883	. . . . . . 7 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎))
152	151	fveq2d 6885	. . . . . 6 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) = ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
153	152	prodeq2dv 15943	. . . . 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 34641	. . . . 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 4147	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑣 = 𝑒) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}) = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
158		vex 3468	. . . . . . . 8 ⊢ 𝑒 ∈ V
159	158, 64	unex 7743	. . . . . . 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 6998	. . . . 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 15744	. . . 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 6883	. . . . . . . . 9 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → (𝑑‘𝑎) = (𝑒‘𝑎))
165	164	fveq2d 6885	. . . . . . . 8 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
166	165	prodeq2dv 15943	. . . . . . 7 ⊢ (𝑑 = 𝑒 → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)))
167	166	oveq1d 7425	. . . . . 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 15717	. . . . 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 2781	. . 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 15723	. 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 2775	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 1540   ∈ wcel 2109  ∀wral 3052  ∃wrex 3061  {crab 3420  Vcvv 3464   ∪ cun 3929   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  {csn 4606  ⟨cop 4612  ∪ ciun 4972   class class class wbr 5124   ↦ cmpt 5206  ran crn 5660   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_m cmap 8845  Fincfn 8964  ℂcc 11132  0cc0 11134  1c1 11135   + caddc 11137   · cmul 11139   ≤ cle 11275   − cmin 11471  ℕcn 12245  ℕ₀cn0 12506  ℤcz 12593  ℤ_≥cuz 12857  ...cfz 13529  ..^cfzo 13676  Σcsu 15707  ∏cprod 15924  reprcrepr 34645

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-cnex 11190  ax-resscn 11191  ax-1cn 11192  ax-icn 11193  ax-addcl 11194  ax-addrcl 11195  ax-mulcl 11196  ax-mulrcl 11197  ax-mulcom 11198  ax-addass 11199  ax-mulass 11200  ax-distr 11201  ax-i2m1 11202  ax-1ne0 11203  ax-1rid 11204  ax-rnegex 11205  ax-rrecex 11206  ax-cnre 11207  ax-pre-lttri 11208  ax-pre-lttrn 11209  ax-pre-ltadd 11210  ax-pre-mulgt0 11211  ax-pre-sup 11212

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-disj 5092  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-pm 8848  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-sup 9459  df-oi 9529  df-card 9958  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-sub 11473  df-neg 11474  df-div 11900  df-nn 12246  df-2 12308  df-3 12309  df-n0 12507  df-z 12594  df-uz 12858  df-rp 13014  df-fz 13530  df-fzo 13677  df-seq 14025  df-exp 14085  df-hash 14354  df-cj 15123  df-re 15124  df-im 15125  df-sqrt 15259  df-abs 15260  df-clim 15509  df-sum 15708  df-prod 15925  df-repr 34646

This theorem is referenced by:  breprexplemc  34669