reprpmtf1o - Mathbox for Thierry Arnoux

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Theorem reprpmtf1o 34994

Description: Transposing 0 and 𝑋 maps representations with a condition on the first index to transpositions with the same condition on the index 𝑋. (Contributed by Thierry Arnoux, 27-Dec-2021.)

**Hypotheses**
Ref	Expression
reprpmtf1o.s	⊢ (𝜑 → 𝑆 ∈ ℕ)
reprpmtf1o.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
reprpmtf1o.a	⊢ (𝜑 → 𝐴 ⊆ ℕ)
reprpmtf1o.x	⊢ (𝜑 → 𝑋 ∈ (0..^𝑆))
reprpmtf1o.o	⊢ 𝑂 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘0) ∈ 𝐵}
reprpmtf1o.p	⊢ 𝑃 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑋) ∈ 𝐵}
reprpmtf1o.t	⊢ 𝑇 = if(𝑋 = 0, ( I ↾ (0..^𝑆)), ((pmTrsp‘(0..^𝑆))‘{𝑋, 0}))
reprpmtf1o.f	⊢ 𝐹 = (𝑐 ∈ 𝑃 ↦ (𝑐 ∘ 𝑇))

**Assertion**
Ref	Expression
reprpmtf1o	⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑂)

Distinct variable groups: 𝐴,𝑐 𝐵,𝑐 𝑀,𝑐 𝑃,𝑐 𝑆,𝑐 𝑇,𝑐 𝑋,𝑐 𝜑,𝑐 Allowed substitution hints: 𝐹(𝑐) 𝑂(𝑐)

Proof of Theorem reprpmtf1o Dummy variables 𝑎 𝑏 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ (𝐴 ↑_m (0..^𝑆)) = (𝐴 ↑_m (0..^𝑆))
2		eqid 2763	. . . . 5 ⊢ (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) = (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇))
3		ovexd 7447	. . . . 5 ⊢ (𝜑 → (0..^𝑆) ∈ V)
4		nnex 12240	. . . . . . 7 ⊢ ℕ ∈ V
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
6		reprpmtf1o.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
7	5, 6	ssexd 5296	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
8		reprpmtf1o.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆))
9		reprpmtf1o.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℕ)
10		lbfzo0 13730	. . . . . . 7 ⊢ (0 ∈ (0..^𝑆) ↔ 𝑆 ∈ ℕ)
11	9, 10	sylibr 237	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0..^𝑆))
12		reprpmtf1o.t	. . . . . 6 ⊢ 𝑇 = if(𝑋 = 0, ( I ↾ (0..^𝑆)), ((pmTrsp‘(0..^𝑆))‘{𝑋, 0}))
13	3, 8, 11, 12	pmtridf1o 33395	. . . . 5 ⊢ (𝜑 → 𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆))
14	1, 1, 2, 3, 3, 7, 13	fmptco1f1o 32959	. . . 4 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))–1-1-onto→(𝐴 ↑_m (0..^𝑆)))
15		f1of1 6821	. . . 4 ⊢ ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))–1-1-onto→(𝐴 ↑_m (0..^𝑆)) → (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))–1-1→(𝐴 ↑_m (0..^𝑆)))
16	14, 15	syl 18	. . 3 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))–1-1→(𝐴 ↑_m (0..^𝑆)))
17		ssrab2 4035	. . . . . 6 ⊢ {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} ⊆ (𝐴 ↑_m (0..^𝑆))
18		reprpmtf1o.p	. . . . . . . . . 10 ⊢ 𝑃 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑋) ∈ 𝐵}
19	18	ssrab3 4037	. . . . . . . . 9 ⊢ 𝑃 ⊆ (𝐴(repr‘𝑆)𝑀)
20	19	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ⊆ (𝐴(repr‘𝑆)𝑀))
21		reprpmtf1o.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
22	9	nnnn0d 12566	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℕ₀)
23	6, 21, 22	reprval 34978	. . . . . . . 8 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
24	20, 23	sseqtrd 3974	. . . . . . 7 ⊢ (𝜑 → 𝑃 ⊆ {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
25	24	sselda 3938	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
26	17, 25	sselid 3936	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ (𝐴 ↑_m (0..^𝑆)))
27	26	ex 417	. . . 4 ⊢ (𝜑 → (𝑐 ∈ 𝑃 → 𝑐 ∈ (𝐴 ↑_m (0..^𝑆))))
28	27	ssrdv 3944	. . 3 ⊢ (𝜑 → 𝑃 ⊆ (𝐴 ↑_m (0..^𝑆)))
29		f1ores 6837	. . 3 ⊢ (((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))–1-1→(𝐴 ↑_m (0..^𝑆)) ∧ 𝑃 ⊆ (𝐴 ↑_m (0..^𝑆))) → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) ↾ 𝑃):𝑃–1-1-onto→((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃))
30	16, 28, 29	syl2anc 595	. 2 ⊢ (𝜑 → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) ↾ 𝑃):𝑃–1-1-onto→((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃))
31		resmpt 6041	. . . . 5 ⊢ (𝑃 ⊆ (𝐴 ↑_m (0..^𝑆)) → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) ↾ 𝑃) = (𝑐 ∈ 𝑃 ↦ (𝑐 ∘ 𝑇)))
32	28, 31	syl 18	. . . 4 ⊢ (𝜑 → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) ↾ 𝑃) = (𝑐 ∈ 𝑃 ↦ (𝑐 ∘ 𝑇)))
33		reprpmtf1o.f	. . . 4 ⊢ 𝐹 = (𝑐 ∈ 𝑃 ↦ (𝑐 ∘ 𝑇))
34	32, 33	eqtr4di 2816	. . 3 ⊢ (𝜑 → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) ↾ 𝑃) = 𝐹)
35		eqidd 2764	. . 3 ⊢ (𝜑 → 𝑃 = 𝑃)
36		vex 3459	. . . . . . . . 9 ⊢ 𝑑 ∈ V
37	36	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑑 ∈ V)
38	2, 37, 28	elimampt 6047	. . . . . . 7 ⊢ (𝜑 → (𝑑 ∈ ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃) ↔ ∃𝑐 ∈ 𝑃 𝑑 = (𝑐 ∘ 𝑇)))
39		simpr 489	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 𝑑 = (𝑐 ∘ 𝑇))
40		f1of 6822	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))–1-1-onto→(𝐴 ↑_m (0..^𝑆)) → (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))⟶(𝐴 ↑_m (0..^𝑆)))
41	14, 40	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))⟶(𝐴 ↑_m (0..^𝑆)))
42	41	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))⟶(𝐴 ↑_m (0..^𝑆)))
43	2	fmpt 7107	. . . . . . . . . . . . . . 15 ⊢ (∀𝑐 ∈ (𝐴 ↑_m (0..^𝑆))(𝑐 ∘ 𝑇) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))⟶(𝐴 ↑_m (0..^𝑆)))
44	42, 43	sylibr 237	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ∀𝑐 ∈ (𝐴 ↑_m (0..^𝑆))(𝑐 ∘ 𝑇) ∈ (𝐴 ↑_m (0..^𝑆)))
45	26	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 𝑐 ∈ (𝐴 ↑_m (0..^𝑆)))
46		rspa 3254	. . . . . . . . . . . . . 14 ⊢ ((∀𝑐 ∈ (𝐴 ↑_m (0..^𝑆))(𝑐 ∘ 𝑇) ∈ (𝐴 ↑_m (0..^𝑆)) ∧ 𝑐 ∈ (𝐴 ↑_m (0..^𝑆))) → (𝑐 ∘ 𝑇) ∈ (𝐴 ↑_m (0..^𝑆)))
47	44, 45, 46	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑐 ∘ 𝑇) ∈ (𝐴 ↑_m (0..^𝑆)))
48	39, 47	eqeltrd 2863	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 𝑑 ∈ (𝐴 ↑_m (0..^𝑆)))
49	39	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑑 = (𝑐 ∘ 𝑇))
50	49	fveq1d 6885	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑑‘𝑎) = ((𝑐 ∘ 𝑇)‘𝑎))
51		f1ofun 6824	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → Fun 𝑇)
52	13, 51	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun 𝑇)
53	52	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → Fun 𝑇)
54		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
55		f1odm 6826	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → dom 𝑇 = (0..^𝑆))
56	13, 55	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → dom 𝑇 = (0..^𝑆))
57	56	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → dom 𝑇 = (0..^𝑆))
58	54, 57	eleqtrrd 2866	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ dom 𝑇)
59		fvco 6981	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝑇 ∧ 𝑎 ∈ dom 𝑇) → ((𝑐 ∘ 𝑇)‘𝑎) = (𝑐‘(𝑇‘𝑎)))
60	53, 58, 59	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑐 ∘ 𝑇)‘𝑎) = (𝑐‘(𝑇‘𝑎)))
61	60	adantlr 727	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑐 ∘ 𝑇)‘𝑎) = (𝑐‘(𝑇‘𝑎)))
62	50, 61	eqtrd 2798	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑑‘𝑎) = (𝑐‘(𝑇‘𝑎)))
63	62	sumeq2dv 15755	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘(𝑇‘𝑎)))
64		fveq2 6883	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑇‘𝑎) → (𝑐‘𝑏) = (𝑐‘(𝑇‘𝑎)))
65		fzofi 14012	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑆) ∈ Fin
66	65	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → (0..^𝑆) ∈ Fin)
67	13	adantr 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆))
68		eqidd 2764	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑇‘𝑎) = (𝑇‘𝑎))
69	6	ad2antrr 738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑏 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
70	6	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝐴 ⊆ ℕ)
71	21	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑀 ∈ ℤ)
72	22	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑆 ∈ ℕ₀)
73	20	sselda 3938	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ (𝐴(repr‘𝑆)𝑀))
74	70, 71, 72, 73	reprf 34980	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐:(0..^𝑆)⟶𝐴)
75	74	ffvelcdmda 7081	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑐‘𝑏) ∈ 𝐴)
76	69, 75	sseldd 3939	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑐‘𝑏) ∈ ℕ)
77	76	nncnd 12250	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑐‘𝑏) ∈ ℂ)
78	64, 66, 67, 68, 77	fsumf1o 15776	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → Σ𝑏 ∈ (0..^𝑆)(𝑐‘𝑏) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘(𝑇‘𝑎)))
79	78	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → Σ𝑏 ∈ (0..^𝑆)(𝑐‘𝑏) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘(𝑇‘𝑎)))
80	70, 71, 72, 73	reprsum 34981	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → Σ𝑏 ∈ (0..^𝑆)(𝑐‘𝑏) = 𝑀)
81	80	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → Σ𝑏 ∈ (0..^𝑆)(𝑐‘𝑏) = 𝑀)
82	63, 79, 81	3eqtr2d 2804	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀)
83		fveq1 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑎) = (𝑑‘𝑎))
84	83	sumeq2sdv 15756	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎))
85	84	eqeq1d 2765	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → (Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀 ↔ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀))
86	85	elrab 3651	. . . . . . . . . . . 12 ⊢ (𝑑 ∈ {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} ↔ (𝑑 ∈ (𝐴 ↑_m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀))
87	48, 82, 86	sylanbrc 594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 𝑑 ∈ {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
88	23	ad2antrr 738	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
89	87, 88	eleqtrrd 2866	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
90	39	fveq1d 6885	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑑‘0) = ((𝑐 ∘ 𝑇)‘0))
91	52	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → Fun 𝑇)
92	11, 56	eleqtrrd 2866	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ dom 𝑇)
93	92	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 0 ∈ dom 𝑇)
94		fvco 6981	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑇 ∧ 0 ∈ dom 𝑇) → ((𝑐 ∘ 𝑇)‘0) = (𝑐‘(𝑇‘0)))
95	91, 93, 94	syl2anc 595	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ((𝑐 ∘ 𝑇)‘0) = (𝑐‘(𝑇‘0)))
96	3, 8, 11, 12	pmtridfv2 33397	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑇‘0) = 𝑋)
97	96	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑇‘0) = 𝑋)
98	97	fveq2d 6887	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑐‘(𝑇‘0)) = (𝑐‘𝑋))
99		simpr 489	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ 𝑃)
100	99, 18	eleqtrdi 2873	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑋) ∈ 𝐵})
101		rabid 3437	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑋) ∈ 𝐵} ↔ (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑐‘𝑋) ∈ 𝐵))
102	100, 101	sylib 221	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑐‘𝑋) ∈ 𝐵))
103	102	simprd 500	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → ¬ (𝑐‘𝑋) ∈ 𝐵)
104	103	adantr 485	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ¬ (𝑐‘𝑋) ∈ 𝐵)
105	98, 104	eqneltrd 2883	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ¬ (𝑐‘(𝑇‘0)) ∈ 𝐵)
106	95, 105	eqneltrd 2883	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ¬ ((𝑐 ∘ 𝑇)‘0) ∈ 𝐵)
107	90, 106	eqneltrd 2883	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ¬ (𝑑‘0) ∈ 𝐵)
108	89, 107	jca 520	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵))
109	108	r19.29an 3169	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑐 ∈ 𝑃 𝑑 = (𝑐 ∘ 𝑇)) → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵))
110	6	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
111	21	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
112	22	adantr 485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ₀)
113		simpr 489	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
114	110, 111, 112, 113	reprf 34980	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
115		f1ocnv 6835	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → ^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆))
116		f1of 6822	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → ^◡𝑇:(0..^𝑆)⟶(0..^𝑆))
117	13, 115, 116	3syl 19	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ^◡𝑇:(0..^𝑆)⟶(0..^𝑆))
118	117	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → ^◡𝑇:(0..^𝑆)⟶(0..^𝑆))
119		fco 6732	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑:(0..^𝑆)⟶𝐴 ∧ ^◡𝑇:(0..^𝑆)⟶(0..^𝑆)) → (𝑑 ∘ ^◡𝑇):(0..^𝑆)⟶𝐴)
120	114, 118, 119	syl2anc 595	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∘ ^◡𝑇):(0..^𝑆)⟶𝐴)
121		elmapg 8837	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ V ∧ (0..^𝑆) ∈ V) → ((𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑑 ∘ ^◡𝑇):(0..^𝑆)⟶𝐴))
122	7, 3, 121	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑑 ∘ ^◡𝑇):(0..^𝑆)⟶𝐴))
123	122	adantr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → ((𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑑 ∘ ^◡𝑇):(0..^𝑆)⟶𝐴))
124	120, 123	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)))
125	124	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)))
126		f1ofun 6824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → Fun ^◡𝑇)
127	13, 115, 126	3syl 19	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Fun ^◡𝑇)
128	127	ad2antrr 738	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑎 ∈ (0..^𝑆)) → Fun ^◡𝑇)
129		simpr 489	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
130		f1odm 6826	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → dom ^◡𝑇 = (0..^𝑆))
131	13, 115, 130	3syl 19	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom ^◡𝑇 = (0..^𝑆))
132	131	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → dom ^◡𝑇 = (0..^𝑆))
133	129, 132	eleqtrrd 2866	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ dom ^◡𝑇)
134	133	adantlr 727	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ dom ^◡𝑇)
135		fvco 6981	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun ^◡𝑇 ∧ 𝑎 ∈ dom ^◡𝑇) → ((𝑑 ∘ ^◡𝑇)‘𝑎) = (𝑑‘(^◡𝑇‘𝑎)))
136	128, 134, 135	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑑 ∘ ^◡𝑇)‘𝑎) = (𝑑‘(^◡𝑇‘𝑎)))
137	136	sumeq2dv 15755	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑑‘(^◡𝑇‘𝑎)))
138		fveq2 6883	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 = (^◡𝑇‘𝑎) → (𝑑‘𝑏) = (𝑑‘(^◡𝑇‘𝑎)))
139	65	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (0..^𝑆) ∈ Fin)
140	13, 115	syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆))
141	140	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → ^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆))
142		eqidd 2764	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑎 ∈ (0..^𝑆)) → (^◡𝑇‘𝑎) = (^◡𝑇‘𝑎))
143	110	adantr 485	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑏 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
144	114	ffvelcdmda 7081	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑑‘𝑏) ∈ 𝐴)
145	143, 144	sseldd 3939	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑑‘𝑏) ∈ ℕ)
146	145	nncnd 12250	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑑‘𝑏) ∈ ℂ)
147	138, 139, 141, 142, 146	fsumf1o 15776	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → Σ𝑏 ∈ (0..^𝑆)(𝑑‘𝑏) = Σ𝑎 ∈ (0..^𝑆)(𝑑‘(^◡𝑇‘𝑎)))
148	110, 111, 112, 113	reprsum 34981	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → Σ𝑏 ∈ (0..^𝑆)(𝑑‘𝑏) = 𝑀)
149	137, 147, 148	3eqtr2d 2804	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎) = 𝑀)
150	149	adantr 485	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎) = 𝑀)
151		fveq1 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → (𝑐‘𝑎) = ((𝑑 ∘ ^◡𝑇)‘𝑎))
152	151	sumeq2sdv 15756	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎))
153	152	eqeq1d 2765	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → (Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀 ↔ Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎) = 𝑀))
154	153	elrab 3651	. . . . . . . . . . . . . 14 ⊢ ((𝑑 ∘ ^◡𝑇) ∈ {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀} ↔ ((𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)) ∧ Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎) = 𝑀))
155	125, 150, 154	sylanbrc 594	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑 ∘ ^◡𝑇) ∈ {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
156	23	ad2antrr 738	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
157	155, 156	eleqtrrd 2866	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑 ∘ ^◡𝑇) ∈ (𝐴(repr‘𝑆)𝑀))
158	127	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → Fun ^◡𝑇)
159	8, 131	eleqtrrd 2866	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ dom ^◡𝑇)
160	159	ad2antrr 738	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → 𝑋 ∈ dom ^◡𝑇)
161		fvco 6981	. . . . . . . . . . . . . 14 ⊢ ((Fun ^◡𝑇 ∧ 𝑋 ∈ dom ^◡𝑇) → ((𝑑 ∘ ^◡𝑇)‘𝑋) = (𝑑‘(^◡𝑇‘𝑋)))
162	158, 160, 161	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → ((𝑑 ∘ ^◡𝑇)‘𝑋) = (𝑑‘(^◡𝑇‘𝑋)))
163		f1ocnvfv 7278	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) ∧ 0 ∈ (0..^𝑆)) → ((𝑇‘0) = 𝑋 → (^◡𝑇‘𝑋) = 0))
164	163	imp 411	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) ∧ 0 ∈ (0..^𝑆)) ∧ (𝑇‘0) = 𝑋) → (^◡𝑇‘𝑋) = 0)
165	13, 11, 96, 164	syl21anc 850	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (^◡𝑇‘𝑋) = 0)
166	165	ad2antrr 738	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (^◡𝑇‘𝑋) = 0)
167	166	fveq2d 6887	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑‘(^◡𝑇‘𝑋)) = (𝑑‘0))
168		simpr 489	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → ¬ (𝑑‘0) ∈ 𝐵)
169	167, 168	eqneltrd 2883	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → ¬ (𝑑‘(^◡𝑇‘𝑋)) ∈ 𝐵)
170	162, 169	eqneltrd 2883	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → ¬ ((𝑑 ∘ ^◡𝑇)‘𝑋) ∈ 𝐵)
171		fveq1 6882	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → (𝑐‘𝑋) = ((𝑑 ∘ ^◡𝑇)‘𝑋))
172	171	eleq1d 2848	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → ((𝑐‘𝑋) ∈ 𝐵 ↔ ((𝑑 ∘ ^◡𝑇)‘𝑋) ∈ 𝐵))
173	172	notbid 321	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → (¬ (𝑐‘𝑋) ∈ 𝐵 ↔ ¬ ((𝑑 ∘ ^◡𝑇)‘𝑋) ∈ 𝐵))
174	173	elrab 3651	. . . . . . . . . . . 12 ⊢ ((𝑑 ∘ ^◡𝑇) ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑋) ∈ 𝐵} ↔ ((𝑑 ∘ ^◡𝑇) ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ ((𝑑 ∘ ^◡𝑇)‘𝑋) ∈ 𝐵))
175	157, 170, 174	sylanbrc 594	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑 ∘ ^◡𝑇) ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑋) ∈ 𝐵})
176	175, 18	eleqtrrdi 2874	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑 ∘ ^◡𝑇) ∈ 𝑃)
177	176	anasss 471	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → (𝑑 ∘ ^◡𝑇) ∈ 𝑃)
178		simpr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) ∧ 𝑐 = (𝑑 ∘ ^◡𝑇)) → 𝑐 = (𝑑 ∘ ^◡𝑇))
179	178	coeq1d 5849	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) ∧ 𝑐 = (𝑑 ∘ ^◡𝑇)) → (𝑐 ∘ 𝑇) = ((𝑑 ∘ ^◡𝑇) ∘ 𝑇))
180	179	eqeq2d 2774	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) ∧ 𝑐 = (𝑑 ∘ ^◡𝑇)) → (𝑑 = (𝑐 ∘ 𝑇) ↔ 𝑑 = ((𝑑 ∘ ^◡𝑇) ∘ 𝑇)))
181		f1ococnv1 6852	. . . . . . . . . . . . . 14 ⊢ (𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → (^◡𝑇 ∘ 𝑇) = ( I ↾ (0..^𝑆)))
182	13, 181	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑇 ∘ 𝑇) = ( I ↾ (0..^𝑆)))
183	182	adantr 485	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → (^◡𝑇 ∘ 𝑇) = ( I ↾ (0..^𝑆)))
184	183	coeq2d 5850	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → (𝑑 ∘ (^◡𝑇 ∘ 𝑇)) = (𝑑 ∘ ( I ↾ (0..^𝑆))))
185	114	adantrr 729	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → 𝑑:(0..^𝑆)⟶𝐴)
186		fcoi1 6754	. . . . . . . . . . . 12 ⊢ (𝑑:(0..^𝑆)⟶𝐴 → (𝑑 ∘ ( I ↾ (0..^𝑆))) = 𝑑)
187	185, 186	syl 18	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → (𝑑 ∘ ( I ↾ (0..^𝑆))) = 𝑑)
188	184, 187	eqtr2d 2799	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → 𝑑 = (𝑑 ∘ (^◡𝑇 ∘ 𝑇)))
189		coass 6269	. . . . . . . . . 10 ⊢ ((𝑑 ∘ ^◡𝑇) ∘ 𝑇) = (𝑑 ∘ (^◡𝑇 ∘ 𝑇))
190	188, 189	eqtr4di 2816	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → 𝑑 = ((𝑑 ∘ ^◡𝑇) ∘ 𝑇))
191	177, 180, 190	rspcedvd 3584	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → ∃𝑐 ∈ 𝑃 𝑑 = (𝑐 ∘ 𝑇))
192	109, 191	impbida 812	. . . . . . 7 ⊢ (𝜑 → (∃𝑐 ∈ 𝑃 𝑑 = (𝑐 ∘ 𝑇) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)))
193	38, 192	bitrd 282	. . . . . 6 ⊢ (𝜑 → (𝑑 ∈ ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)))
194		fveq1 6882	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → (𝑐‘0) = (𝑑‘0))
195	194	eleq1d 2848	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → ((𝑐‘0) ∈ 𝐵 ↔ (𝑑‘0) ∈ 𝐵))
196	195	notbid 321	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘0) ∈ 𝐵 ↔ ¬ (𝑑‘0) ∈ 𝐵))
197	196	elrab 3651	. . . . . 6 ⊢ (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘0) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵))
198	193, 197	bitr4di 292	. . . . 5 ⊢ (𝜑 → (𝑑 ∈ ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃) ↔ 𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘0) ∈ 𝐵}))
199	198	eqrdv 2761	. . . 4 ⊢ (𝜑 → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃) = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘0) ∈ 𝐵})
200		reprpmtf1o.o	. . . 4 ⊢ 𝑂 = {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘0) ∈ 𝐵}
201	199, 200	eqtr4di 2816	. . 3 ⊢ (𝜑 → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃) = 𝑂)
202	34, 35, 201	f1oeq123d 6816	. 2 ⊢ (𝜑 → (((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) ↾ 𝑃):𝑃–1-1-onto→((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃) ↔ 𝐹:𝑃–1-1-onto→𝑂))
203	30, 202	mpbid 235	1 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑂)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∀wral 3079 ∃wrex 3089 {crab 3416 Vcvv 3455 ⊆ wss 3906 ifcif 4488 {cpr 4592 ↦ cmpt 5193 I cid 5557 ^◡ccnv 5662 dom cdm 5663 ↾ cres 5665 “ cima 5666 ∘ ccom 5667 Fun wfun 6532 ⟶wf 6534 –1-1→wf1 6535 –1-1-onto→wf1o 6537 ‘cfv 6538 (class class class)co 7412 ↑_m cmap 8825 Fincfn 8944 0cc0 11101 ℕcn 12234 ℕ₀cn0 12505 ℤcz 12592 ..^cfzo 13684 Σcsu 15739 pmTrspcpmtr 19512 reprcrepr 34976

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7864 df-1st 7987 df-2nd 7988 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-er 8695 df-map 8827 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-sup 9403 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-n0 12506 df-z 12593 df-uz 12864 df-rp 13018 df-fz 13537 df-fzo 13685 df-seq 14040 df-exp 14100 df-hash 14369 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-clim 15541 df-sum 15740 df-pmtr 19513 df-repr 34977

This theorem is referenced by: hgt750lema 35025

W3C validator