Theorem breprexplema 34646

Description: Lemma for breprexp 34649 (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 13596	. . . . 5 ⊢ (1...𝑁) ⊆ ℕ
2	1	a1i 11	. . . 4 ⊢ (𝜑 → (1...𝑁) ⊆ ℕ)
3		breprexplema.m	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℕ0)
4	3	nn0zd 12641	. . . 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 34631	. . 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 15737	. 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 14015	. . 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 13567	. . . . . . . 8 ⊢ (1...𝑁) ⊆ ℤ
13		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ (1...𝑁))
14	12, 13	sselid 3980	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → 𝑏 ∈ ℤ)
15	11, 14	zsubcld 12729	. . . . . 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 34632	. . . . 5 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ∈ Fin)
19		mptfi 9392	. . . . 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 9381	. . . 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 34626	. . . . 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 4079	. . . . 5 ⊢ {𝑐 ∈ ((1...𝑁) ↑m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = (𝑀 − 𝑏)} ⊆ ((1...𝑁) ↑m (0..^𝑆))
25	23, 24	eqsstrdi 4027	. . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ (1...𝑁)) → ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ⊆ ((1...𝑁) ↑m (0..^𝑆)))
26	9	elexd 3503	. . . 4 ⊢ (𝜑 → (1...𝑁) ∈ V)
27		fzonel 13714	. . . . 5 ⊢ ¬ 𝑆 ∈ (0..^𝑆)
28	27	a1i 11	. . . 4 ⊢ (𝜑 → ¬ 𝑆 ∈ (0..^𝑆))
29	25, 26, 5, 28, 6	actfunsnrndisj 34621	. . 3 ⊢ (𝜑 → Disj 𝑏 ∈ (1...𝑁)ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩})))
30		fzofi 14016	. . . . . 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 3145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (0..^(𝑆 + 1))) → ∀𝑦 ∈ ℕ ((𝐿‘𝑥)‘𝑦) ∈ ℂ)
34	33	ralrimiva 3145	. . . . . . 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 1913	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣(𝜑 ∧ 𝑏 ∈ (1...𝑁))
38		nfcv 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣𝑑
39		nfmpt1 5249	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑣(𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
40	39	nfrn 5962	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑣ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
41	38, 40	nfel 2919	. . . . . . . . . . . 12 ⊢ Ⅎ𝑣 𝑑 ∈ ran (𝑣 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑣 ∪ {⟨𝑆, 𝑏⟩}))
42	37, 41	nfan 1898	. . . . . . . . . . 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 34628	. . . . . . . . . . . . 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 6890	. . . . . . . . . . . . 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 13738	. . . . . . . . . . . . . 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 6770	. . . . . . . . . . . . 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 12921	. . . . . . . . . . . . . . . 16 ⊢ ℕ0 = (ℤ≥‘0)
56	5, 55	eleqtrdi 2850	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 ∈ (ℤ≥‘0))
57		fzosplitsn 13815	. . . . . . . . . . . . . . 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 6723	. . . . . . . . . . . 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 3483	. . . . . . . . . . . . . 14 ⊢ 𝑣 ∈ V
64		snex 5435	. . . . . . . . . . . . . 14 ⊢ {⟨𝑆, 𝑏⟩} ∈ V
65	63, 64	unex 7765	. . . . . . . . . . . . 13 ⊢ (𝑣 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
66	6, 65	elrnmpti 5972	. . . . . . . . . . . 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 3267	. . . . . . . . . 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 7104	. . . . . . . 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 3980	. . . . . . 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 6905	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (𝐿‘𝑥) = (𝐿‘𝑎))
73	72	fveq1d 6907	. . . . . . . . 9 ⊢ (𝑥 = 𝑎 → ((𝐿‘𝑥)‘𝑦) = ((𝐿‘𝑎)‘𝑦))
74	73	eleq1d 2825	. . . . . . . 8 ⊢ (𝑥 = 𝑎 → (((𝐿‘𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘𝑦) ∈ ℂ))
75		fveq2 6905	. . . . . . . . 9 ⊢ (𝑦 = (𝑑‘𝑎) → ((𝐿‘𝑎)‘𝑦) = ((𝐿‘𝑎)‘(𝑑‘𝑎)))
76	75	eleq1d 2825	. . . . . . . 8 ⊢ (𝑦 = (𝑑‘𝑎) → (((𝐿‘𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘(𝑑‘𝑎)) ∈ ℂ))
77	74, 76	rspc2v 3632	. . . . . . 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 15989	. . . 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 15858	. 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 15957	. . . . . 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 1913	. . . . . . 7 ⊢ Ⅎ𝑎((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏)))
86		nfcv 2904	. . . . . . 7 ⊢ Ⅎ𝑎((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
87		fzofi 14016	. . . . . . . 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 34628	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑒:(0..^𝑆)⟶(1...𝑁))
95	94	ffnd 6736	. . . . . . . . . . 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 6617	. . . . . . . . . . . 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 6999	. . . . . . . . . 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 1375	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑒‘𝑎))
105	104	fveq2d 6909	. . . . . . . 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 13783	. . . . . . . . . . . . 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 3982	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
112	94	ffvelcdmda 7103	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ (1...𝑁))
113	1, 112	sselid 3980	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ ℕ)
114		fveq2 6905	. . . . . . . . . . . 12 ⊢ (𝑦 = (𝑒‘𝑎) → ((𝐿‘𝑎)‘𝑦) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
115	114	eleq1d 2825	. . . . . . . . . . 11 ⊢ (𝑦 = (𝑒‘𝑎) → (((𝐿‘𝑎)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑎)‘(𝑒‘𝑎)) ∈ ℂ))
116	74, 115	rspc2v 3632	. . . . . . . . . 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 6905	. . . . . . . 8 ⊢ (𝑎 = 𝑆 → (𝐿‘𝑎) = (𝐿‘𝑆))
121		fveq2 6905	. . . . . . . 8 ⊢ (𝑎 = 𝑆 → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
122	120, 121	fveq12d 6912	. . . . . . 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 4659	. . . . . . . . . . . 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 7000	. . . . . . . . . . 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 1375	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
128		fvsng 7201	. . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = 𝑏)
131	130	fveq2d 6909	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → ((𝐿‘𝑆)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑆)) = ((𝐿‘𝑆)‘𝑏))
132		fzonn0p1 13782	. . . . . . . . . . . 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 3980	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) → 𝑏 ∈ ℕ)
136		fveq2 6905	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑆 → (𝐿‘𝑥) = (𝐿‘𝑆))
137	136	fveq1d 6907	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑆 → ((𝐿‘𝑥)‘𝑦) = ((𝐿‘𝑆)‘𝑦))
138	137	eleq1d 2825	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑆 → (((𝐿‘𝑥)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑆)‘𝑦) ∈ ℂ))
139		fveq2 6905	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑏 → ((𝐿‘𝑆)‘𝑦) = ((𝐿‘𝑆)‘𝑏))
140	139	eleq1d 2825	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → (((𝐿‘𝑆)‘𝑦) ∈ ℂ ↔ ((𝐿‘𝑆)‘𝑏) ∈ ℂ))
141	138, 140	rspc2v 3632	. . . . . . . . . 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 16026	. . . . . 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 15959	. . . . . . 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 7450	. . . . . 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 2780	. . . . 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 15739	. . . 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 6907	. . . . . . 7 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → (𝑑‘𝑎) = ((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎))
152	151	fveq2d 6909	. . . . . 6 ⊢ ((𝑑 = (𝑒 ∪ {⟨𝑆, 𝑏⟩}) ∧ 𝑎 ∈ (0..^(𝑆 + 1))) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) = ((𝐿‘𝑎)‘((𝑒 ∪ {⟨𝑆, 𝑏⟩})‘𝑎)))
153	152	prodeq2dv 15959	. . . . 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 34620	. . . . 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 4166	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑏 ∈ (1...𝑁)) ∧ 𝑒 ∈ ((1...𝑁)(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑣 = 𝑒) → (𝑣 ∪ {⟨𝑆, 𝑏⟩}) = (𝑒 ∪ {⟨𝑆, 𝑏⟩}))
158		vex 3483	. . . . . . . 8 ⊢ 𝑒 ∈ V
159	158, 64	unex 7765	. . . . . . 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 7022	. . . . 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 15760	. . . 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 6907	. . . . . . . . 9 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → (𝑑‘𝑎) = (𝑒‘𝑎))
165	164	fveq2d 6909	. . . . . . . 8 ⊢ ((𝑑 = 𝑒 ∧ 𝑎 ∈ (0..^𝑆)) → ((𝐿‘𝑎)‘(𝑑‘𝑎)) = ((𝐿‘𝑎)‘(𝑒‘𝑎)))
166	165	prodeq2dv 15959	. . . . . . 7 ⊢ (𝑑 = 𝑒 → ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑑‘𝑎)) = ∏𝑎 ∈ (0..^𝑆)((𝐿‘𝑎)‘(𝑒‘𝑎)))
167	166	oveq1d 7447	. . . . . 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 15733	. . . . 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 2786	. . 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 15739	. 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 2780	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 1539   ∈ wcel 2107  ∀wral 3060  ∃wrex 3069  {crab 3435  Vcvv 3479   ∪ cun 3948   ∩ cin 3949   ⊆ wss 3950  ∅c0 4332  {csn 4625  ⟨cop 4631  ∪ ciun 4990   class class class wbr 5142   ↦ cmpt 5224  ran crn 5685   Fn wfn 6555  ⟶wf 6556  ‘cfv 6560  (class class class)co 7432   ↑_m cmap 8867  Fincfn 8986  ℂcc 11154  0cc0 11156  1c1 11157   + caddc 11159   · cmul 11161   ≤ cle 11297   − cmin 11493  ℕcn 12267  ℕ₀cn0 12528  ℤcz 12615  ℤ_≥cuz 12879  ...cfz 13548  ..^cfzo 13695  Σcsu 15723  ∏cprod 15940  reprcrepr 34624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-inf2 9682  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-disj 5110  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-er 8746  df-map 8869  df-pm 8870  df-en 8987  df-dom 8988  df-sdom 8989  df-fin 8990  df-sup 9483  df-oi 9551  df-card 9980  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-3 12331  df-n0 12529  df-z 12616  df-uz 12880  df-rp 13036  df-fz 13549  df-fzo 13696  df-seq 14044  df-exp 14104  df-hash 14371  df-cj 15139  df-re 15140  df-im 15141  df-sqrt 15275  df-abs 15276  df-clim 15525  df-sum 15724  df-prod 15941  df-repr 34625

This theorem is referenced by:  breprexplemc  34648