reprsuc - Mathbox for Thierry Arnoux

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Theorem reprsuc 34799

Description: Express the representations recursively. (Contributed by Thierry Arnoux, 5-Dec-2021.)

**Hypotheses**
Ref	Expression
reprval.a	⊢ (𝜑 → 𝐴 ⊆ ℕ)
reprval.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
reprval.s	⊢ (𝜑 → 𝑆 ∈ ℕ₀)
reprsuc.f	⊢ 𝐹 = (𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑐 ∪ {⟨𝑆, 𝑏⟩}))

**Assertion**
Ref	Expression
reprsuc	⊢ (𝜑 → (𝐴(repr‘(𝑆 + 1))𝑀) = ∪ 𝑏 ∈ 𝐴 ran 𝐹)

Distinct variable groups: 𝐴,𝑏,𝑐 𝑀,𝑏,𝑐 𝑆,𝑏,𝑐 𝜑,𝑏,𝑐 Allowed substitution hints: 𝐹(𝑏,𝑐)

Proof of Theorem reprsuc Dummy variables 𝑎 𝑑 𝑒 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reprval.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
2		reprval.m	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		reprval.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ ℕ₀)
4		1nn0 12444	. . . . 5 ⊢ 1 ∈ ℕ₀
5	4	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ ℕ₀)
6	3, 5	nn0addcld 12493	. . 3 ⊢ (𝜑 → (𝑆 + 1) ∈ ℕ₀)
7	1, 2, 6	reprval 34794	. 2 ⊢ (𝜑 → (𝐴(repr‘(𝑆 + 1))𝑀) = {𝑐 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∣ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = 𝑀})
8		simplr 774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))))
9		elmapi 8786	. . . . . . . . . 10 ⊢ (𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) → 𝑒:(0..^(𝑆 + 1))⟶𝐴)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑒:(0..^(𝑆 + 1))⟶𝐴)
11	3	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑆 ∈ ℕ₀)
12		fzonn0p1 13688	. . . . . . . . . 10 ⊢ (𝑆 ∈ ℕ₀ → 𝑆 ∈ (0..^(𝑆 + 1)))
13	11, 12	syl 17	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑆 ∈ (0..^(𝑆 + 1)))
14	10, 13	ffvelcdmd 7026	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒‘𝑆) ∈ 𝐴)
15		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ∧ 𝑏 = (𝑒‘𝑆)) → 𝑏 = (𝑒‘𝑆))
16	15	oveq2d 7372	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ∧ 𝑏 = (𝑒‘𝑆)) → (𝑀 − 𝑏) = (𝑀 − (𝑒‘𝑆)))
17	16	oveq2d 7372	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ∧ 𝑏 = (𝑒‘𝑆)) → (𝐴(repr‘𝑆)(𝑀 − 𝑏)) = (𝐴(repr‘𝑆)(𝑀 − (𝑒‘𝑆))))
18		opeq2 4805	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑒‘𝑆) → ⟨𝑆, 𝑏⟩ = ⟨𝑆, (𝑒‘𝑆)⟩)
19	18	sneqd 4567	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑒‘𝑆) → {⟨𝑆, 𝑏⟩} = {⟨𝑆, (𝑒‘𝑆)⟩})
20	19	uneq2d 4098	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑒‘𝑆) → (𝑐 ∪ {⟨𝑆, 𝑏⟩}) = (𝑐 ∪ {⟨𝑆, (𝑒‘𝑆)⟩}))
21	20	eqeq2d 2750	. . . . . . . . . 10 ⊢ (𝑏 = (𝑒‘𝑆) → (𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}) ↔ 𝑒 = (𝑐 ∪ {⟨𝑆, (𝑒‘𝑆)⟩})))
22	21	adantl 482	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ∧ 𝑏 = (𝑒‘𝑆)) → (𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}) ↔ 𝑒 = (𝑐 ∪ {⟨𝑆, (𝑒‘𝑆)⟩})))
23	17, 22	rexeqbidv 3314	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ∧ 𝑏 = (𝑒‘𝑆)) → (∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}) ↔ ∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − (𝑒‘𝑆)))𝑒 = (𝑐 ∪ {⟨𝑆, (𝑒‘𝑆)⟩})))
24	9	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → 𝑒:(0..^(𝑆 + 1))⟶𝐴)
25	3	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → 𝑆 ∈ ℕ₀)
26		fzossfzop1 13689	. . . . . . . . . . . . . . . 16 ⊢ (𝑆 ∈ ℕ₀ → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
27	25, 26	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → (0..^𝑆) ⊆ (0..^(𝑆 + 1)))
28	24, 27	fssresd 6694	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → (𝑒 ↾ (0..^𝑆)):(0..^𝑆)⟶𝐴)
29	28	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒 ↾ (0..^𝑆)):(0..^𝑆)⟶𝐴)
30		nnex 12171	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ∈ V
31	30	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℕ ∈ V)
32	31, 1	ssexd 5252	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ V)
33		fzofi 13927	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑆) ∈ Fin
34	33	elexi 3453	. . . . . . . . . . . . . . 15 ⊢ (0..^𝑆) ∈ V
35		elmapg 8776	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ V ∧ (0..^𝑆) ∈ V) → ((𝑒 ↾ (0..^𝑆)) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑒 ↾ (0..^𝑆)):(0..^𝑆)⟶𝐴))
36	32, 34, 35	sylancl 592	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑒 ↾ (0..^𝑆)) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑒 ↾ (0..^𝑆)):(0..^𝑆)⟶𝐴))
37	36	ad2antrr 732	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → ((𝑒 ↾ (0..^𝑆)) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑒 ↾ (0..^𝑆)):(0..^𝑆)⟶𝐴))
38	29, 37	mpbird 258	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒 ↾ (0..^𝑆)) ∈ (𝐴 ↑_m (0..^𝑆)))
39	33	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → (0..^𝑆) ∈ Fin)
40		nnsscn 12170	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ ⊆ ℂ
41	40	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ℕ ⊆ ℂ)
42	1, 41	sstrd 3925	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
43	42	ad2antrr 732	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℂ)
44	28	ffvelcdmda 7025	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ↾ (0..^𝑆))‘𝑎) ∈ 𝐴)
45	43, 44	sseldd 3916	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ↾ (0..^𝑆))‘𝑎) ∈ ℂ)
46	39, 45	fsumcl 15686	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) ∈ ℂ)
47	46	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) ∈ ℂ)
48	42	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → 𝐴 ⊆ ℂ)
49	25, 12	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → 𝑆 ∈ (0..^(𝑆 + 1)))
50	24, 49	ffvelcdmd 7026	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → (𝑒‘𝑆) ∈ 𝐴)
51	48, 50	sseldd 3916	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → (𝑒‘𝑆) ∈ ℂ)
52	51	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒‘𝑆) ∈ ℂ)
53	47, 52	pncand 11497	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → ((Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) + (𝑒‘𝑆)) − (𝑒‘𝑆)) = Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎))
54		nfv 1921	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))))
55		nfcv 2901	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎(𝑒‘𝑆)
56		fzonel 13619	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 𝑆 ∈ (0..^𝑆)
57	56	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → ¬ 𝑆 ∈ (0..^𝑆))
58	24	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒:(0..^(𝑆 + 1))⟶𝐴)
59	27	sselda 3915	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
60	58, 59	ffvelcdmd 7026	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ 𝐴)
61	43, 60	sseldd 3916	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ ℂ)
62		fveq2 6827	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑆 → (𝑒‘𝑎) = (𝑒‘𝑆))
63	54, 55, 39, 25, 57, 61, 62, 51	fsumsplitsn 15697	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → Σ𝑎 ∈ ((0..^𝑆) ∪ {𝑆})(𝑒‘𝑎) = (Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) + (𝑒‘𝑆)))
64		fzosplitsn 13722	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ (ℤ_≥‘0) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
65		nn0uz 12817	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ₀ = (ℤ_≥‘0)
66	64, 65	eleq2s 2857	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆 ∈ ℕ₀ → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
67	25, 66	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
68	67	sumeq1d 15653	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = Σ𝑎 ∈ ((0..^𝑆) ∪ {𝑆})(𝑒‘𝑎))
69		simpr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
70	69	fvresd 6847	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ↾ (0..^𝑆))‘𝑎) = (𝑒‘𝑎))
71	70	sumeq2dv 15655	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎))
72	71	oveq1d 7371	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → (Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) + (𝑒‘𝑆)) = (Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) + (𝑒‘𝑆)))
73	63, 68, 72	3eqtr4d 2784	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = (Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) + (𝑒‘𝑆)))
74	73	adantr 481	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = (Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) + (𝑒‘𝑆)))
75		simpr 485	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀)
76	74, 75	eqtr3d 2776	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) + (𝑒‘𝑆)) = 𝑀)
77	76	oveq1d 7371	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → ((Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) + (𝑒‘𝑆)) − (𝑒‘𝑆)) = (𝑀 − (𝑒‘𝑆)))
78	53, 77	eqtr3d 2776	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) = (𝑀 − (𝑒‘𝑆)))
79	38, 78	jca 516	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → ((𝑒 ↾ (0..^𝑆)) ∈ (𝐴 ↑_m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) = (𝑀 − (𝑒‘𝑆))))
80		fveq1 6826	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝑒 ↾ (0..^𝑆)) → (𝑑‘𝑎) = ((𝑒 ↾ (0..^𝑆))‘𝑎))
81	80	sumeq2sdv 15656	. . . . . . . . . . . . 13 ⊢ (𝑑 = (𝑒 ↾ (0..^𝑆)) → Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎))
82	81	eqeq1d 2741	. . . . . . . . . . . 12 ⊢ (𝑑 = (𝑒 ↾ (0..^𝑆)) → (Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = (𝑀 − (𝑒‘𝑆)) ↔ Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) = (𝑀 − (𝑒‘𝑆))))
83	82	elrab 3629	. . . . . . . . . . 11 ⊢ ((𝑒 ↾ (0..^𝑆)) ∈ {𝑑 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = (𝑀 − (𝑒‘𝑆))} ↔ ((𝑒 ↾ (0..^𝑆)) ∈ (𝐴 ↑_m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) = (𝑀 − (𝑒‘𝑆))))
84	79, 83	sylibr 235	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒 ↾ (0..^𝑆)) ∈ {𝑑 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = (𝑀 − (𝑒‘𝑆))})
85	1	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝐴 ⊆ ℕ)
86	2	ad2antrr 732	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑀 ∈ ℤ)
87		nnssz 12537	. . . . . . . . . . . . . . 15 ⊢ ℕ ⊆ ℤ
88	1, 87	sstrdi 3927	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
89	88	ad2antrr 732	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝐴 ⊆ ℤ)
90	89, 14	sseldd 3916	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒‘𝑆) ∈ ℤ)
91	86, 90	zsubcld 12629	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑀 − (𝑒‘𝑆)) ∈ ℤ)
92	85, 91, 11	reprval 34794	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝐴(repr‘𝑆)(𝑀 − (𝑒‘𝑆))) = {𝑑 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = (𝑀 − (𝑒‘𝑆))})
93	84, 92	eleqtrrd 2842	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒 ↾ (0..^𝑆)) ∈ (𝐴(repr‘𝑆)(𝑀 − (𝑒‘𝑆))))
94		simpr 485	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ∧ 𝑐 = (𝑒 ↾ (0..^𝑆))) → 𝑐 = (𝑒 ↾ (0..^𝑆)))
95	94	uneq1d 4097	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ∧ 𝑐 = (𝑒 ↾ (0..^𝑆))) → (𝑐 ∪ {⟨𝑆, (𝑒‘𝑆)⟩}) = ((𝑒 ↾ (0..^𝑆)) ∪ {⟨𝑆, (𝑒‘𝑆)⟩}))
96	95	eqeq2d 2750	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ∧ 𝑐 = (𝑒 ↾ (0..^𝑆))) → (𝑒 = (𝑐 ∪ {⟨𝑆, (𝑒‘𝑆)⟩}) ↔ 𝑒 = ((𝑒 ↾ (0..^𝑆)) ∪ {⟨𝑆, (𝑒‘𝑆)⟩})))
97	10	ffnd 6656	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑒 Fn (0..^(𝑆 + 1)))
98		fnsnsplit 7128	. . . . . . . . . . 11 ⊢ ((𝑒 Fn (0..^(𝑆 + 1)) ∧ 𝑆 ∈ (0..^(𝑆 + 1))) → 𝑒 = ((𝑒 ↾ ((0..^(𝑆 + 1)) ∖ {𝑆})) ∪ {⟨𝑆, (𝑒‘𝑆)⟩}))
99	97, 13, 98	syl2anc 590	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑒 = ((𝑒 ↾ ((0..^(𝑆 + 1)) ∖ {𝑆})) ∪ {⟨𝑆, (𝑒‘𝑆)⟩}))
100	11, 65	eleqtrdi 2849	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑆 ∈ (ℤ_≥‘0))
101		fzodif2 32883	. . . . . . . . . . . . 13 ⊢ (𝑆 ∈ (ℤ_≥‘0) → ((0..^(𝑆 + 1)) ∖ {𝑆}) = (0..^𝑆))
102	100, 101	syl 17	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → ((0..^(𝑆 + 1)) ∖ {𝑆}) = (0..^𝑆))
103	102	reseq2d 5931	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒 ↾ ((0..^(𝑆 + 1)) ∖ {𝑆})) = (𝑒 ↾ (0..^𝑆)))
104	103	uneq1d 4097	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → ((𝑒 ↾ ((0..^(𝑆 + 1)) ∖ {𝑆})) ∪ {⟨𝑆, (𝑒‘𝑆)⟩}) = ((𝑒 ↾ (0..^𝑆)) ∪ {⟨𝑆, (𝑒‘𝑆)⟩}))
105	99, 104	eqtrd 2774	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑒 = ((𝑒 ↾ (0..^𝑆)) ∪ {⟨𝑆, (𝑒‘𝑆)⟩}))
106	93, 96, 105	rspcedvd 3562	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → ∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − (𝑒‘𝑆)))𝑒 = (𝑐 ∪ {⟨𝑆, (𝑒‘𝑆)⟩}))
107	14, 23, 106	rspcedvd 3562	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
108	107	anasss 467	. . . . . 6 ⊢ ((𝜑 ∧ (𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀)) → ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
109		simpr 485	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
110	1	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝐴 ⊆ ℕ)
111	110	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝐴 ⊆ ℕ)
112	2	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑀 ∈ ℤ)
113	88	sselda 3915	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ ℤ)
114	112, 113	zsubcld 12629	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (𝑀 − 𝑏) ∈ ℤ)
115	114	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → (𝑀 − 𝑏) ∈ ℤ)
116	3	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑆 ∈ ℕ₀)
117	116	adantr 481	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝑆 ∈ ℕ₀)
118		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏)))
119	111, 115, 117, 118	reprf 34796	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝑐:(0..^𝑆)⟶𝐴)
120		simplr 774	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝑏 ∈ 𝐴)
121	117, 120	fsnd 6811	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐴)
122		fzodisjsn 13643	. . . . . . . . . . . . . 14 ⊢ ((0..^𝑆) ∩ {𝑆}) = ∅
123	122	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → ((0..^𝑆) ∩ {𝑆}) = ∅)
124	119, 121, 123	fun2d 6691	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → (𝑐 ∪ {⟨𝑆, 𝑏⟩}):((0..^𝑆) ∪ {𝑆})⟶𝐴)
125	117, 66	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
126	125	feq2d 6639	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑐 ∪ {⟨𝑆, 𝑏⟩}):(0..^(𝑆 + 1))⟶𝐴 ↔ (𝑐 ∪ {⟨𝑆, 𝑏⟩}):((0..^𝑆) ∪ {𝑆})⟶𝐴))
127	124, 126	mpbird 258	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → (𝑐 ∪ {⟨𝑆, 𝑏⟩}):(0..^(𝑆 + 1))⟶𝐴)
128		ovex 7389	. . . . . . . . . . . . 13 ⊢ (0..^(𝑆 + 1)) ∈ V
129		elmapg 8776	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ V ∧ (0..^(𝑆 + 1)) ∈ V) → ((𝑐 ∪ {⟨𝑆, 𝑏⟩}) ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ↔ (𝑐 ∪ {⟨𝑆, 𝑏⟩}):(0..^(𝑆 + 1))⟶𝐴))
130	32, 128, 129	sylancl 592	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑐 ∪ {⟨𝑆, 𝑏⟩}) ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ↔ (𝑐 ∪ {⟨𝑆, 𝑏⟩}):(0..^(𝑆 + 1))⟶𝐴))
131	130	ad2antrr 732	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → ((𝑐 ∪ {⟨𝑆, 𝑏⟩}) ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ↔ (𝑐 ∪ {⟨𝑆, 𝑏⟩}):(0..^(𝑆 + 1))⟶𝐴))
132	127, 131	mpbird 258	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → (𝑐 ∪ {⟨𝑆, 𝑏⟩}) ∈ (𝐴 ↑_m (0..^(𝑆 + 1))))
133	132	adantr 481	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑐 ∪ {⟨𝑆, 𝑏⟩}) ∈ (𝐴 ↑_m (0..^(𝑆 + 1))))
134	109, 133	eqeltrd 2839	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))))
135	125	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
136	135	sumeq1d 15653	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = Σ𝑎 ∈ ((0..^𝑆) ∪ {𝑆})(𝑒‘𝑎))
137		nfv 1921	. . . . . . . . . 10 ⊢ Ⅎ𝑎(((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
138	33	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (0..^𝑆) ∈ Fin)
139	117	adantr 481	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑆 ∈ ℕ₀)
140	56	a1i 11	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → ¬ 𝑆 ∈ (0..^𝑆))
141	42	ad4antr 738	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℂ)
142	127	adantr 481	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑐 ∪ {⟨𝑆, 𝑏⟩}):(0..^(𝑆 + 1))⟶𝐴)
143	109	feq1d 6637	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑒:(0..^(𝑆 + 1))⟶𝐴 ↔ (𝑐 ∪ {⟨𝑆, 𝑏⟩}):(0..^(𝑆 + 1))⟶𝐴))
144	142, 143	mpbird 258	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑒:(0..^(𝑆 + 1))⟶𝐴)
145	144	adantr 481	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒:(0..^(𝑆 + 1))⟶𝐴)
146		simpr 485	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
147		elun1 4111	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ (0..^𝑆) → 𝑎 ∈ ((0..^𝑆) ∪ {𝑆}))
148	146, 147	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ ((0..^𝑆) ∪ {𝑆}))
149	125	ad2antrr 732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
150	148, 149	eleqtrrd 2842	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
151	145, 150	ffvelcdmd 7026	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ 𝐴)
152	141, 151	sseldd 3916	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ ℂ)
153	42	ad3antrrr 736	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝐴 ⊆ ℂ)
154	139, 12	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑆 ∈ (0..^(𝑆 + 1)))
155	144, 154	ffvelcdmd 7026	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑒‘𝑆) ∈ 𝐴)
156	153, 155	sseldd 3916	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑒‘𝑆) ∈ ℂ)
157	137, 55, 138, 139, 140, 152, 62, 156	fsumsplitsn 15697	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → Σ𝑎 ∈ ((0..^𝑆) ∪ {𝑆})(𝑒‘𝑎) = (Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) + (𝑒‘𝑆)))
158		simplr 774	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
159	158	fveq1d 6829	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) = ((𝑐 ∪ {⟨𝑆, 𝑏⟩})‘𝑎))
160	119	ffnd 6656	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝑐 Fn (0..^𝑆))
161	160	ad2antrr 732	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑐 Fn (0..^𝑆))
162	121	ffnd 6656	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
163	162	ad2antrr 732	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
164	122	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → ((0..^𝑆) ∩ {𝑆}) = ∅)
165		fvun1 6918	. . . . . . . . . . . . . . . 16 ⊢ ((𝑐 Fn (0..^𝑆) ∧ {⟨𝑆, 𝑏⟩} Fn {𝑆} ∧ (((0..^𝑆) ∩ {𝑆}) = ∅ ∧ 𝑎 ∈ (0..^𝑆))) → ((𝑐 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑐‘𝑎))
166	161, 163, 164, 146, 165	syl112anc 1382	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑐 ∪ {⟨𝑆, 𝑏⟩})‘𝑎) = (𝑐‘𝑎))
167	159, 166	eqtrd 2774	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) = (𝑐‘𝑎))
168	167	ralrimiva 3131	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → ∀𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) = (𝑐‘𝑎))
169	168	sumeq2d 15654	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎))
170	111	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝐴 ⊆ ℕ)
171	115	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑀 − 𝑏) ∈ ℤ)
172	118	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏)))
173	170, 171, 139, 172	reprsum 34797	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = (𝑀 − 𝑏))
174	169, 173	eqtrd 2774	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) = (𝑀 − 𝑏))
175	109	fveq1d 6829	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑒‘𝑆) = ((𝑐 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
176	160	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑐 Fn (0..^𝑆))
177	162	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
178	122	a1i 11	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → ((0..^𝑆) ∩ {𝑆}) = ∅)
179		snidg 4592	. . . . . . . . . . . . . 14 ⊢ (𝑆 ∈ ℕ₀ → 𝑆 ∈ {𝑆})
180	139, 179	syl 17	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑆 ∈ {𝑆})
181		fvun2 6919	. . . . . . . . . . . . 13 ⊢ ((𝑐 Fn (0..^𝑆) ∧ {⟨𝑆, 𝑏⟩} Fn {𝑆} ∧ (((0..^𝑆) ∩ {𝑆}) = ∅ ∧ 𝑆 ∈ {𝑆})) → ((𝑐 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
182	176, 177, 178, 180, 181	syl112anc 1382	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → ((𝑐 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
183	120	adantr 481	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑏 ∈ 𝐴)
184		fvsng 7124	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ ℕ₀ ∧ 𝑏 ∈ 𝐴) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
185	139, 183, 184	syl2anc 590	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
186	175, 182, 185	3eqtrd 2778	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑒‘𝑆) = 𝑏)
187	174, 186	oveq12d 7374	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) + (𝑒‘𝑆)) = ((𝑀 − 𝑏) + 𝑏))
188		zsscn 12523	. . . . . . . . . . . 12 ⊢ ℤ ⊆ ℂ
189	112	ad2antrr 732	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑀 ∈ ℤ)
190	188, 189	sselid 3913	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑀 ∈ ℂ)
191	186, 156	eqeltrrd 2840	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑏 ∈ ℂ)
192	190, 191	npcand 11500	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → ((𝑀 − 𝑏) + 𝑏) = 𝑀)
193	187, 192	eqtrd 2774	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) + (𝑒‘𝑆)) = 𝑀)
194	136, 157, 193	3eqtrd 2778	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀)
195	134, 194	jca 516	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀))
196	195	r19.29ffa 32558	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀))
197	108, 196	impbida 806	. . . . 5 ⊢ (𝜑 → ((𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ↔ ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})))
198		reprsuc.f	. . . . . . 7 ⊢ 𝐹 = (𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏)) ↦ (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
199		vex 3435	. . . . . . . 8 ⊢ 𝑐 ∈ V
200		snex 5368	. . . . . . . 8 ⊢ {⟨𝑆, 𝑏⟩} ∈ V
201	199, 200	unex 7687	. . . . . . 7 ⊢ (𝑐 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
202	198, 201	elrnmpti 5904	. . . . . 6 ⊢ (𝑒 ∈ ran 𝐹 ↔ ∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
203	202	rexbii 3086	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 𝑒 ∈ ran 𝐹 ↔ ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
204	197, 203	bitr4di 290	. . . 4 ⊢ (𝜑 → ((𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ↔ ∃𝑏 ∈ 𝐴 𝑒 ∈ ran 𝐹))
205		fveq1 6826	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐‘𝑎) = (𝑒‘𝑎))
206	205	sumeq2sdv 15656	. . . . . . 7 ⊢ (𝑐 = 𝑒 → Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎))
207	206	eqeq1d 2741	. . . . . 6 ⊢ (𝑐 = 𝑒 → (Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = 𝑀 ↔ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀))
208	207	cbvrabv 3401	. . . . 5 ⊢ {𝑐 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∣ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = 𝑀} = {𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∣ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀}
209	208	reqabi 3414	. . . 4 ⊢ (𝑒 ∈ {𝑐 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∣ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = 𝑀} ↔ (𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀))
210		eliun 4925	. . . 4 ⊢ (𝑒 ∈ ∪ 𝑏 ∈ 𝐴 ran 𝐹 ↔ ∃𝑏 ∈ 𝐴 𝑒 ∈ ran 𝐹)
211	204, 209, 210	3bitr4g 315	. . 3 ⊢ (𝜑 → (𝑒 ∈ {𝑐 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∣ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = 𝑀} ↔ 𝑒 ∈ ∪ 𝑏 ∈ 𝐴 ran 𝐹))
212	211	eqrdv 2737	. 2 ⊢ (𝜑 → {𝑐 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∣ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = 𝑀} = ∪ 𝑏 ∈ 𝐴 ran 𝐹)
213	7, 212	eqtrd 2774	1 ⊢ (𝜑 → (𝐴(repr‘(𝑆 + 1))𝑀) = ∪ 𝑏 ∈ 𝐴 ran 𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ∃wrex 3063 {crab 3391 Vcvv 3431 ∖ cdif 3880 ∪ cun 3881 ∩ cin 3882 ⊆ wss 3883 ∅c0 4261 {csn 4555 ⟨cop 4561 ∪ ciun 4921 ↦ cmpt 5153 ran crn 5619 ↾ cres 5620 Fn wfn 6480 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ↑_m cmap 8763 Fincfn 8883 ℂcc 11027 0cc0 11029 1c1 11030 + caddc 11032 − cmin 11368 ℕcn 12165 ℕ₀cn0 12428 ℤcz 12515 ℤ_≥cuz 12779 ..^cfzo 13599 Σcsu 15639 reprcrepr 34792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-1st 7931 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-n0 12429 df-z 12516 df-uz 12780 df-rp 12934 df-fz 13453 df-fzo 13600 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15441 df-sum 15640 df-repr 34793

This theorem is referenced by: breprexplema 34814

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