reprsuc - Mathbox for Thierry Arnoux

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Theorem reprsuc 31881

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 11907	. . . . 5 ⊢ 1 ∈ ℕ₀
5	4	a1i 11	. . . 4 ⊢ (𝜑 → 1 ∈ ℕ₀)
6	3, 5	nn0addcld 11953	. . 3 ⊢ (𝜑 → (𝑆 + 1) ∈ ℕ₀)
7	1, 2, 6	reprval 31876	. 2 ⊢ (𝜑 → (𝐴(repr‘(𝑆 + 1))𝑀) = {𝑐 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∣ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = 𝑀})
8		simplr 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))))
9		elmapi 8422	. . . . . . . . . 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 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑆 ∈ ℕ₀)
12		fzonn0p1 13108	. . . . . . . . . 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	ffvelrnd 6847	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒‘𝑆) ∈ 𝐴)
15		simpr 487	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ∧ 𝑏 = (𝑒‘𝑆)) → 𝑏 = (𝑒‘𝑆))
16	15	oveq2d 7166	. . . . . . . . . 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 7166	. . . . . . . . 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 4798	. . . . . . . . . . . . 13 ⊢ (𝑏 = (𝑒‘𝑆) → ⟨𝑆, 𝑏⟩ = ⟨𝑆, (𝑒‘𝑆)⟩)
19	18	sneqd 4573	. . . . . . . . . . . 12 ⊢ (𝑏 = (𝑒‘𝑆) → {⟨𝑆, 𝑏⟩} = {⟨𝑆, (𝑒‘𝑆)⟩})
20	19	uneq2d 4139	. . . . . . . . . . 11 ⊢ (𝑏 = (𝑒‘𝑆) → (𝑐 ∪ {⟨𝑆, 𝑏⟩}) = (𝑐 ∪ {⟨𝑆, (𝑒‘𝑆)⟩}))
21	20	eqeq2d 2832	. . . . . . . . . 10 ⊢ (𝑏 = (𝑒‘𝑆) → (𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}) ↔ 𝑒 = (𝑐 ∪ {⟨𝑆, (𝑒‘𝑆)⟩})))
22	21	adantl 484	. . . . . . . . 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 3403	. . . . . . . 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 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → 𝑒:(0..^(𝑆 + 1))⟶𝐴)
25	3	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → 𝑆 ∈ ℕ₀)
26		fzossfzop1 13109	. . . . . . . . . . . . . . . 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 6540	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → (𝑒 ↾ (0..^𝑆)):(0..^𝑆)⟶𝐴)
29	28	adantr 483	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒 ↾ (0..^𝑆)):(0..^𝑆)⟶𝐴)
30		nnex 11638	. . . . . . . . . . . . . . . . 17 ⊢ ℕ ∈ V
31	30	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℕ ∈ V)
32	31, 1	ssexd 5221	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ∈ V)
33		fzofi 13336	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑆) ∈ Fin
34	33	elexi 3514	. . . . . . . . . . . . . . 15 ⊢ (0..^𝑆) ∈ V
35		elmapg 8413	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ V ∧ (0..^𝑆) ∈ V) → ((𝑒 ↾ (0..^𝑆)) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑒 ↾ (0..^𝑆)):(0..^𝑆)⟶𝐴))
36	32, 34, 35	sylancl 588	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑒 ↾ (0..^𝑆)) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑒 ↾ (0..^𝑆)):(0..^𝑆)⟶𝐴))
37	36	ad2antrr 724	. . . . . . . . . . . . 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 259	. . . . . . . . . . . 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 11637	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ ⊆ ℂ
41	40	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ℕ ⊆ ℂ)
42	1, 41	sstrd 3977	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
43	42	ad2antrr 724	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℂ)
44	28	ffvelrnda 6846	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ↾ (0..^𝑆))‘𝑎) ∈ 𝐴)
45	43, 44	sseldd 3968	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ↾ (0..^𝑆))‘𝑎) ∈ ℂ)
46	39, 45	fsumcl 15084	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) ∈ ℂ)
47	46	adantr 483	. . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 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	ffvelrnd 6847	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → (𝑒‘𝑆) ∈ 𝐴)
51	48, 50	sseldd 3968	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → (𝑒‘𝑆) ∈ ℂ)
52	51	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒‘𝑆) ∈ ℂ)
53	47, 52	pncand 10992	. . . . . . . . . . . . 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 1911	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))))
55		nfcv 2977	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑎(𝑒‘𝑆)
56		fzonel 13045	. . . . . . . . . . . . . . . . . . 19 ⊢ ¬ 𝑆 ∈ (0..^𝑆)
57	56	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → ¬ 𝑆 ∈ (0..^𝑆))
58	24	adantr 483	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒:(0..^(𝑆 + 1))⟶𝐴)
59	27	sselda 3967	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
60	58, 59	ffvelrnd 6847	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ 𝐴)
61	43, 60	sseldd 3968	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ ℂ)
62		fveq2 6665	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 = 𝑆 → (𝑒‘𝑎) = (𝑒‘𝑆))
63	54, 55, 39, 25, 57, 61, 62, 51	fsumsplitsn 15094	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → Σ𝑎 ∈ ((0..^𝑆) ∪ {𝑆})(𝑒‘𝑎) = (Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) + (𝑒‘𝑆)))
64		fzosplitsn 13139	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑆 ∈ (ℤ_≥‘0) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
65		nn0uz 12274	. . . . . . . . . . . . . . . . . . . 20 ⊢ ℕ₀ = (ℤ_≥‘0)
66	64, 65	eleq2s 2931	. . . . . . . . . . . . . . . . . . 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 15052	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = Σ𝑎 ∈ ((0..^𝑆) ∪ {𝑆})(𝑒‘𝑎))
69		simpr 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
70	69	fvresd 6685	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑒 ↾ (0..^𝑆))‘𝑎) = (𝑒‘𝑎))
71	70	sumeq2dv 15054	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) → Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎))
72	71	oveq1d 7165	. . . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . 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 487	. . . . . . . . . . . . . . 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 2858	. . . . . . . . . . . . . 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 7165	. . . . . . . . . . . . 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 2858	. . . . . . . . . . . 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 514	. . . . . . . . . . 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 6664	. . . . . . . . . . . . . 14 ⊢ (𝑑 = (𝑒 ↾ (0..^𝑆)) → (𝑑‘𝑎) = ((𝑒 ↾ (0..^𝑆))‘𝑎))
81	80	sumeq2sdv 15055	. . . . . . . . . . . . 13 ⊢ (𝑑 = (𝑒 ↾ (0..^𝑆)) → Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = Σ𝑎 ∈ (0..^𝑆)((𝑒 ↾ (0..^𝑆))‘𝑎))
82	81	eqeq1d 2823	. . . . . . . . . . . 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 3680	. . . . . . . . . . 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 236	. . . . . . . . . 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 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝐴 ⊆ ℕ)
86	2	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑀 ∈ ℤ)
87		nnssz 11996	. . . . . . . . . . . . . . 15 ⊢ ℕ ⊆ ℤ
88	1, 87	sstrdi 3979	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ℤ)
89	88	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝐴 ⊆ ℤ)
90	89, 14	sseldd 3968	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑒‘𝑆) ∈ ℤ)
91	86, 90	zsubcld 12086	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → (𝑀 − (𝑒‘𝑆)) ∈ ℤ)
92	85, 91, 11	reprval 31876	. . . . . . . . . 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 2916	. . . . . . . . 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 487	. . . . . . . . . . 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 4138	. . . . . . . . . 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 2832	. . . . . . . . 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 6510	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑒 Fn (0..^(𝑆 + 1)))
98		fnsnsplit 6941	. . . . . . . . . . 11 ⊢ ((𝑒 Fn (0..^(𝑆 + 1)) ∧ 𝑆 ∈ (0..^(𝑆 + 1))) → 𝑒 = ((𝑒 ↾ ((0..^(𝑆 + 1)) ∖ {𝑆})) ∪ {⟨𝑆, (𝑒‘𝑆)⟩}))
99	97, 13, 98	syl2anc 586	. . . . . . . . . 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 2923	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1)))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) → 𝑆 ∈ (ℤ_≥‘0))
101		fzodif2 30509	. . . . . . . . . . . . 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 5848	. . . . . . . . . . 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 4138	. . . . . . . . . 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 2856	. . . . . . . . 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 3626	. . . . . . . 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 3626	. . . . . . 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 469	. . . . . 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 487	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
110	1	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝐴 ⊆ ℕ)
111	110	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝐴 ⊆ ℕ)
112	2	adantr 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑀 ∈ ℤ)
113	88	sselda 3967	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑏 ∈ ℤ)
114	112, 113	zsubcld 12086	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → (𝑀 − 𝑏) ∈ ℤ)
115	114	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → (𝑀 − 𝑏) ∈ ℤ)
116	3	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑏 ∈ 𝐴) → 𝑆 ∈ ℕ₀)
117	116	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝑆 ∈ ℕ₀)
118		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏)))
119	111, 115, 117, 118	reprf 31878	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝑐:(0..^𝑆)⟶𝐴)
120		simplr 767	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝑏 ∈ 𝐴)
121	117, 120	fsnd 6652	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → {⟨𝑆, 𝑏⟩}:{𝑆}⟶𝐴)
122		fzodisjsn 13069	. . . . . . . . . . . . . 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 6537	. . . . . . . . . . . 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 6495	. . . . . . . . . . . 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 259	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → (𝑐 ∪ {⟨𝑆, 𝑏⟩}):(0..^(𝑆 + 1))⟶𝐴)
128		ovex 7183	. . . . . . . . . . . . 13 ⊢ (0..^(𝑆 + 1)) ∈ V
129		elmapg 8413	. . . . . . . . . . . . 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 588	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑐 ∪ {⟨𝑆, 𝑏⟩}) ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ↔ (𝑐 ∪ {⟨𝑆, 𝑏⟩}):(0..^(𝑆 + 1))⟶𝐴))
131	130	ad2antrr 724	. . . . . . . . . . 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 259	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → (𝑐 ∪ {⟨𝑆, 𝑏⟩}) ∈ (𝐴 ↑_m (0..^(𝑆 + 1))))
133	132	adantr 483	. . . . . . . . 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 2913	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))))
135	125	adantr 483	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (0..^(𝑆 + 1)) = ((0..^𝑆) ∪ {𝑆}))
136	135	sumeq1d 15052	. . . . . . . . 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 1911	. . . . . . . . . 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 483	. . . . . . . . . 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 730	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝐴 ⊆ ℂ)
142	127	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑐 ∪ {⟨𝑆, 𝑏⟩}):(0..^(𝑆 + 1))⟶𝐴)
143	109	feq1d 6494	. . . . . . . . . . . . . 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 259	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑒:(0..^(𝑆 + 1))⟶𝐴)
145	144	adantr 483	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒:(0..^(𝑆 + 1))⟶𝐴)
146		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
147		elun1 4152	. . . . . . . . . . . . . 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 724	. . . . . . . . . . . . 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 2916	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^(𝑆 + 1)))
151	145, 150	ffvelrnd 6847	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ 𝐴)
152	141, 151	sseldd 3968	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) ∈ ℂ)
153	42	ad3antrrr 728	. . . . . . . . . . 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	ffvelrnd 6847	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑒‘𝑆) ∈ 𝐴)
156	153, 155	sseldd 3968	. . . . . . . . . 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 15094	. . . . . . . . 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 767	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
159	158	fveq1d 6667	. . . . . . . . . . . . . . 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 6510	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → 𝑐 Fn (0..^𝑆))
161	160	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑐 Fn (0..^𝑆))
162	121	ffnd 6510	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) → {⟨𝑆, 𝑏⟩} Fn {𝑆})
163	162	ad2antrr 724	. . . . . . . . . . . . . . . 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 6749	. . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑒‘𝑎) = (𝑐‘𝑎))
168	167	ralrimiva 3182	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → ∀𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) = (𝑐‘𝑎))
169	168	sumeq2d 15053	. . . . . . . . . . . 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 483	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝐴 ⊆ ℕ)
171	115	adantr 483	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑀 − 𝑏) ∈ ℤ)
172	118	adantr 483	. . . . . . . . . . . . 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 31879	. . . . . . . . . . . 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 2856	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → Σ𝑎 ∈ (0..^𝑆)(𝑒‘𝑎) = (𝑀 − 𝑏))
175	109	fveq1d 6667	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑒‘𝑆) = ((𝑐 ∪ {⟨𝑆, 𝑏⟩})‘𝑆))
176	160	adantr 483	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑐 Fn (0..^𝑆))
177	162	adantr 483	. . . . . . . . . . . . 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 4593	. . . . . . . . . . . . . 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 6750	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → ((𝑐 ∪ {⟨𝑆, 𝑏⟩})‘𝑆) = ({⟨𝑆, 𝑏⟩}‘𝑆))
183	120	adantr 483	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑏 ∈ 𝐴)
184		fvsng 6937	. . . . . . . . . . . . 13 ⊢ ((𝑆 ∈ ℕ₀ ∧ 𝑏 ∈ 𝐴) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
185	139, 183, 184	syl2anc 586	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → ({⟨𝑆, 𝑏⟩}‘𝑆) = 𝑏)
186	175, 182, 185	3eqtrd 2860	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → (𝑒‘𝑆) = 𝑏)
187	174, 186	oveq12d 7168	. . . . . . . . . 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 11983	. . . . . . . . . . . 12 ⊢ ℤ ⊆ ℂ
189	112	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑀 ∈ ℤ)
190	188, 189	sseldi 3965	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑀 ∈ ℂ)
191	186, 156	eqeltrrd 2914	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → 𝑏 ∈ ℂ)
192	190, 191	npcand 10995	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑏 ∈ 𝐴) ∧ 𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))) ∧ 𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩})) → ((𝑀 − 𝑏) + 𝑏) = 𝑀)
193	187, 192	eqtrd 2856	. . . . . . . . 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 2860	. . . . . . . 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 514	. . . . . . 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 30231	. . . . . 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 799	. . . . 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 3498	. . . . . . . 8 ⊢ 𝑐 ∈ V
200		snex 5324	. . . . . . . 8 ⊢ {⟨𝑆, 𝑏⟩} ∈ V
201	199, 200	unex 7463	. . . . . . 7 ⊢ (𝑐 ∪ {⟨𝑆, 𝑏⟩}) ∈ V
202	198, 201	elrnmpti 5827	. . . . . 6 ⊢ (𝑒 ∈ ran 𝐹 ↔ ∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
203	202	rexbii 3247	. . . . 5 ⊢ (∃𝑏 ∈ 𝐴 𝑒 ∈ ran 𝐹 ↔ ∃𝑏 ∈ 𝐴 ∃𝑐 ∈ (𝐴(repr‘𝑆)(𝑀 − 𝑏))𝑒 = (𝑐 ∪ {⟨𝑆, 𝑏⟩}))
204	197, 203	syl6bbr 291	. . . 4 ⊢ (𝜑 → ((𝑒 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∧ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀) ↔ ∃𝑏 ∈ 𝐴 𝑒 ∈ ran 𝐹))
205		fveq1 6664	. . . . . . . 8 ⊢ (𝑐 = 𝑒 → (𝑐‘𝑎) = (𝑒‘𝑎))
206	205	sumeq2sdv 15055	. . . . . . 7 ⊢ (𝑐 = 𝑒 → Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎))
207	206	eqeq1d 2823	. . . . . 6 ⊢ (𝑐 = 𝑒 → (Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = 𝑀 ↔ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑒‘𝑎) = 𝑀))
208	207	cbvrabv 3492	. . . . 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	rabeq2i 3488	. . . 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 4916	. . . 4 ⊢ (𝑒 ∈ ∪ 𝑏 ∈ 𝐴 ran 𝐹 ↔ ∃𝑏 ∈ 𝐴 𝑒 ∈ ran 𝐹)
211	204, 209, 210	3bitr4g 316	. . 3 ⊢ (𝜑 → (𝑒 ∈ {𝑐 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∣ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = 𝑀} ↔ 𝑒 ∈ ∪ 𝑏 ∈ 𝐴 ran 𝐹))
212	211	eqrdv 2819	. 2 ⊢ (𝜑 → {𝑐 ∈ (𝐴 ↑_m (0..^(𝑆 + 1))) ∣ Σ𝑎 ∈ (0..^(𝑆 + 1))(𝑐‘𝑎) = 𝑀} = ∪ 𝑏 ∈ 𝐴 ran 𝐹)
213	7, 212	eqtrd 2856	1 ⊢ (𝜑 → (𝐴(repr‘(𝑆 + 1))𝑀) = ∪ 𝑏 ∈ 𝐴 ran 𝐹)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∃wrex 3139 {crab 3142 Vcvv 3495 ∖ cdif 3933 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 {csn 4561 ⟨cop 4567 ∪ ciun 4912 ↦ cmpt 5139 ran crn 5551 ↾ cres 5552 Fn wfn 6345 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ↑_m cmap 8400 Fincfn 8503 ℂcc 10529 0cc0 10531 1c1 10532 + caddc 10534 − cmin 10864 ℕcn 11632 ℕ₀cn0 11891 ℤcz 11975 ℤ_≥cuz 12237 ..^cfzo 13027 Σcsu 15036 reprcrepr 31874

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-sum 15037 df-repr 31875

This theorem is referenced by: breprexplema 31896

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