reprpmtf1o - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > reprpmtf1o		Structured version Visualization version GIF version

Theorem reprpmtf1o 34880

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 2761	. . . . 5 ⊢ (𝐴 ↑_m (0..^𝑆)) = (𝐴 ↑_m (0..^𝑆))
2		eqid 2761	. . . . 5 ⊢ (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) = (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇))
3		ovexd 7425	. . . . 5 ⊢ (𝜑 → (0..^𝑆) ∈ V)
4		nnex 12209	. . . . . . 7 ⊢ ℕ ∈ V
5	4	a1i 11	. . . . . 6 ⊢ (𝜑 → ℕ ∈ V)
6		reprpmtf1o.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ⊆ ℕ)
7	5, 6	ssexd 5277	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
8		reprpmtf1o.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (0..^𝑆))
9		reprpmtf1o.s	. . . . . . 7 ⊢ (𝜑 → 𝑆 ∈ ℕ)
10		lbfzo0 13698	. . . . . . 7 ⊢ (0 ∈ (0..^𝑆) ↔ 𝑆 ∈ ℕ)
11	9, 10	sylibr 236	. . . . . 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 33234	. . . . 5 ⊢ (𝜑 → 𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆))
14	1, 1, 2, 3, 3, 7, 13	fmptco1f1o 32795	. . . 4 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))–1-1-onto→(𝐴 ↑_m (0..^𝑆)))
15		f1of1 6799	. . . 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 17	. . 3 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))–1-1→(𝐴 ↑_m (0..^𝑆)))
17		ssrab2 4031	. . . . . 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 4033	. . . . . . . . 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 12535	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ ℕ₀)
23	6, 21, 22	reprval 34864	. . . . . . . 8 ⊢ (𝜑 → (𝐴(repr‘𝑆)𝑀) = {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
24	20, 23	sseqtrd 3970	. . . . . . 7 ⊢ (𝜑 → 𝑃 ⊆ {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
25	24	sselda 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
26	17, 25	sselid 3932	. . . . 5 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ (𝐴 ↑_m (0..^𝑆)))
27	26	ex 416	. . . 4 ⊢ (𝜑 → (𝑐 ∈ 𝑃 → 𝑐 ∈ (𝐴 ↑_m (0..^𝑆))))
28	27	ssrdv 3940	. . 3 ⊢ (𝜑 → 𝑃 ⊆ (𝐴 ↑_m (0..^𝑆)))
29		f1ores 6815	. . 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 593	. 2 ⊢ (𝜑 → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) ↾ 𝑃):𝑃–1-1-onto→((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃))
31		resmpt 6021	. . . . 5 ⊢ (𝑃 ⊆ (𝐴 ↑_m (0..^𝑆)) → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) ↾ 𝑃) = (𝑐 ∈ 𝑃 ↦ (𝑐 ∘ 𝑇)))
32	28, 31	syl 17	. . . 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 2814	. . 3 ⊢ (𝜑 → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) ↾ 𝑃) = 𝐹)
35		eqidd 2762	. . 3 ⊢ (𝜑 → 𝑃 = 𝑃)
36		vex 3457	. . . . . . . . 9 ⊢ 𝑑 ∈ V
37	36	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑑 ∈ V)
38	2, 37, 28	elimampt 6027	. . . . . . 7 ⊢ (𝜑 → (𝑑 ∈ ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃) ↔ ∃𝑐 ∈ 𝑃 𝑑 = (𝑐 ∘ 𝑇)))
39		simpr 488	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 𝑑 = (𝑐 ∘ 𝑇))
40		f1of 6800	. . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)):(𝐴 ↑_m (0..^𝑆))⟶(𝐴 ↑_m (0..^𝑆)))
42	41	ad2antrr 736	. . . . . . . . . . . . . . 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 7085	. . . . . . . . . . . . . . 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 236	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ∀𝑐 ∈ (𝐴 ↑_m (0..^𝑆))(𝑐 ∘ 𝑇) ∈ (𝐴 ↑_m (0..^𝑆)))
45	26	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 𝑐 ∈ (𝐴 ↑_m (0..^𝑆)))
46		rspa 3250	. . . . . . . . . . . . . 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 593	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑐 ∘ 𝑇) ∈ (𝐴 ↑_m (0..^𝑆)))
48	39, 47	eqeltrd 2861	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 𝑑 ∈ (𝐴 ↑_m (0..^𝑆)))
49	39	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑑 = (𝑐 ∘ 𝑇))
50	49	fveq1d 6863	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑑‘𝑎) = ((𝑐 ∘ 𝑇)‘𝑎))
51		f1ofun 6802	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → Fun 𝑇)
52	13, 51	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → Fun 𝑇)
53	52	ad2antrr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → Fun 𝑇)
54		simpr 488	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
55		f1odm 6804	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → dom 𝑇 = (0..^𝑆))
56	13, 55	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → dom 𝑇 = (0..^𝑆))
57	56	ad2antrr 736	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → dom 𝑇 = (0..^𝑆))
58	54, 57	eleqtrrd 2864	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ dom 𝑇)
59		fvco 6959	. . . . . . . . . . . . . . . . 17 ⊢ ((Fun 𝑇 ∧ 𝑎 ∈ dom 𝑇) → ((𝑐 ∘ 𝑇)‘𝑎) = (𝑐‘(𝑇‘𝑎)))
60	53, 58, 59	syl2anc 593	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑐 ∘ 𝑇)‘𝑎) = (𝑐‘(𝑇‘𝑎)))
61	60	adantlr 725	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑐 ∘ 𝑇)‘𝑎) = (𝑐‘(𝑇‘𝑎)))
62	50, 61	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑑‘𝑎) = (𝑐‘(𝑇‘𝑎)))
63	62	sumeq2dv 15719	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘(𝑇‘𝑎)))
64		fveq2 6861	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑇‘𝑎) → (𝑐‘𝑏) = (𝑐‘(𝑇‘𝑎)))
65		fzofi 13980	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝑆) ∈ Fin
66	65	a1i 11	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → (0..^𝑆) ∈ Fin)
67	13	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆))
68		eqidd 2762	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑎 ∈ (0..^𝑆)) → (𝑇‘𝑎) = (𝑇‘𝑎))
69	6	ad2antrr 736	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑏 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
70	6	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝐴 ⊆ ℕ)
71	21	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑀 ∈ ℤ)
72	22	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑆 ∈ ℕ₀)
73	20	sselda 3934	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ (𝐴(repr‘𝑆)𝑀))
74	70, 71, 72, 73	reprf 34866	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐:(0..^𝑆)⟶𝐴)
75	74	ffvelcdmda 7059	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑐‘𝑏) ∈ 𝐴)
76	69, 75	sseldd 3935	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑐‘𝑏) ∈ ℕ)
77	76	nncnd 12219	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑐‘𝑏) ∈ ℂ)
78	64, 66, 67, 68, 77	fsumf1o 15740	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → Σ𝑏 ∈ (0..^𝑆)(𝑐‘𝑏) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘(𝑇‘𝑎)))
79	78	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → Σ𝑏 ∈ (0..^𝑆)(𝑐‘𝑏) = Σ𝑎 ∈ (0..^𝑆)(𝑐‘(𝑇‘𝑎)))
80	70, 71, 72, 73	reprsum 34867	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → Σ𝑏 ∈ (0..^𝑆)(𝑐‘𝑏) = 𝑀)
81	80	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → Σ𝑏 ∈ (0..^𝑆)(𝑐‘𝑏) = 𝑀)
82	63, 79, 81	3eqtr2d 2802	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀)
83		fveq1 6860	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑑 → (𝑐‘𝑎) = (𝑑‘𝑎))
84	83	sumeq2sdv 15720	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎))
85	84	eqeq1d 2763	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → (Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀 ↔ Σ𝑎 ∈ (0..^𝑆)(𝑑‘𝑎) = 𝑀))
86	85	elrab 3649	. . . . . . . . . . . 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 592	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 𝑑 ∈ {𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ∣ Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀})
88	23	ad2antrr 736	. . . . . . . . . . 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 2864	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
90	39	fveq1d 6863	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑑‘0) = ((𝑐 ∘ 𝑇)‘0))
91	52	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → Fun 𝑇)
92	11, 56	eleqtrrd 2864	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 ∈ dom 𝑇)
93	92	ad2antrr 736	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → 0 ∈ dom 𝑇)
94		fvco 6959	. . . . . . . . . . . . 13 ⊢ ((Fun 𝑇 ∧ 0 ∈ dom 𝑇) → ((𝑐 ∘ 𝑇)‘0) = (𝑐‘(𝑇‘0)))
95	91, 93, 94	syl2anc 593	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ((𝑐 ∘ 𝑇)‘0) = (𝑐‘(𝑇‘0)))
96	3, 8, 11, 12	pmtridfv2 33236	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑇‘0) = 𝑋)
97	96	ad2antrr 736	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑇‘0) = 𝑋)
98	97	fveq2d 6865	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑐‘(𝑇‘0)) = (𝑐‘𝑋))
99		simpr 488	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ 𝑃)
100	99, 18	eleqtrdi 2871	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → 𝑐 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑋) ∈ 𝐵})
101		rabid 3434	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑋) ∈ 𝐵} ↔ (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑐‘𝑋) ∈ 𝐵))
102	100, 101	sylib 220	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → (𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑐‘𝑋) ∈ 𝐵))
103	102	simprd 499	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑐 ∈ 𝑃) → ¬ (𝑐‘𝑋) ∈ 𝐵)
104	103	adantr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ¬ (𝑐‘𝑋) ∈ 𝐵)
105	98, 104	eqneltrd 2881	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ¬ (𝑐‘(𝑇‘0)) ∈ 𝐵)
106	95, 105	eqneltrd 2881	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ¬ ((𝑐 ∘ 𝑇)‘0) ∈ 𝐵)
107	90, 106	eqneltrd 2881	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → ¬ (𝑑‘0) ∈ 𝐵)
108	89, 107	jca 519	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑐 ∈ 𝑃) ∧ 𝑑 = (𝑐 ∘ 𝑇)) → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵))
109	108	r19.29an 3165	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑐 ∈ 𝑃 𝑑 = (𝑐 ∘ 𝑇)) → (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵))
110	6	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝐴 ⊆ ℕ)
111	21	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑀 ∈ ℤ)
112	22	adantr 484	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑆 ∈ ℕ₀)
113		simpr 488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑 ∈ (𝐴(repr‘𝑆)𝑀))
114	110, 111, 112, 113	reprf 34866	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → 𝑑:(0..^𝑆)⟶𝐴)
115		f1ocnv 6813	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → ^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆))
116		f1of 6800	. . . . . . . . . . . . . . . . . . 19 ⊢ (^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → ^◡𝑇:(0..^𝑆)⟶(0..^𝑆))
117	13, 115, 116	3syl 18	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ^◡𝑇:(0..^𝑆)⟶(0..^𝑆))
118	117	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → ^◡𝑇:(0..^𝑆)⟶(0..^𝑆))
119		fco 6710	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑑:(0..^𝑆)⟶𝐴 ∧ ^◡𝑇:(0..^𝑆)⟶(0..^𝑆)) → (𝑑 ∘ ^◡𝑇):(0..^𝑆)⟶𝐴)
120	114, 118, 119	syl2anc 593	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∘ ^◡𝑇):(0..^𝑆)⟶𝐴)
121		elmapg 8813	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ V ∧ (0..^𝑆) ∈ V) → ((𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑑 ∘ ^◡𝑇):(0..^𝑆)⟶𝐴))
122	7, 3, 121	syl2anc 593	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑑 ∘ ^◡𝑇):(0..^𝑆)⟶𝐴))
123	122	adantr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → ((𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)) ↔ (𝑑 ∘ ^◡𝑇):(0..^𝑆)⟶𝐴))
124	120, 123	mpbird 259	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → (𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)))
125	124	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑 ∘ ^◡𝑇) ∈ (𝐴 ↑_m (0..^𝑆)))
126		f1ofun 6802	. . . . . . . . . . . . . . . . . . . 20 ⊢ (^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → Fun ^◡𝑇)
127	13, 115, 126	3syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → Fun ^◡𝑇)
128	127	ad2antrr 736	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑎 ∈ (0..^𝑆)) → Fun ^◡𝑇)
129		simpr 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ (0..^𝑆))
130		f1odm 6804	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → dom ^◡𝑇 = (0..^𝑆))
131	13, 115, 130	3syl 18	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → dom ^◡𝑇 = (0..^𝑆))
132	131	adantr 484	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → dom ^◡𝑇 = (0..^𝑆))
133	129, 132	eleqtrrd 2864	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ dom ^◡𝑇)
134	133	adantlr 725	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑎 ∈ (0..^𝑆)) → 𝑎 ∈ dom ^◡𝑇)
135		fvco 6959	. . . . . . . . . . . . . . . . . 18 ⊢ ((Fun ^◡𝑇 ∧ 𝑎 ∈ dom ^◡𝑇) → ((𝑑 ∘ ^◡𝑇)‘𝑎) = (𝑑‘(^◡𝑇‘𝑎)))
136	128, 134, 135	syl2anc 593	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑎 ∈ (0..^𝑆)) → ((𝑑 ∘ ^◡𝑇)‘𝑎) = (𝑑‘(^◡𝑇‘𝑎)))
137	136	sumeq2dv 15719	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎) = Σ𝑎 ∈ (0..^𝑆)(𝑑‘(^◡𝑇‘𝑎)))
138		fveq2 6861	. . . . . . . . . . . . . . . . 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 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆))
141	140	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → ^◡𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆))
142		eqidd 2762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑎 ∈ (0..^𝑆)) → (^◡𝑇‘𝑎) = (^◡𝑇‘𝑎))
143	110	adantr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑏 ∈ (0..^𝑆)) → 𝐴 ⊆ ℕ)
144	114	ffvelcdmda 7059	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑑‘𝑏) ∈ 𝐴)
145	143, 144	sseldd 3935	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑑‘𝑏) ∈ ℕ)
146	145	nncnd 12219	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ 𝑏 ∈ (0..^𝑆)) → (𝑑‘𝑏) ∈ ℂ)
147	138, 139, 141, 142, 146	fsumf1o 15740	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → Σ𝑏 ∈ (0..^𝑆)(𝑑‘𝑏) = Σ𝑎 ∈ (0..^𝑆)(𝑑‘(^◡𝑇‘𝑎)))
148	110, 111, 112, 113	reprsum 34867	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → Σ𝑏 ∈ (0..^𝑆)(𝑑‘𝑏) = 𝑀)
149	137, 147, 148	3eqtr2d 2802	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) → Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎) = 𝑀)
150	149	adantr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎) = 𝑀)
151		fveq1 6860	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → (𝑐‘𝑎) = ((𝑑 ∘ ^◡𝑇)‘𝑎))
152	151	sumeq2sdv 15720	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎))
153	152	eqeq1d 2763	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → (Σ𝑎 ∈ (0..^𝑆)(𝑐‘𝑎) = 𝑀 ↔ Σ𝑎 ∈ (0..^𝑆)((𝑑 ∘ ^◡𝑇)‘𝑎) = 𝑀))
154	153	elrab 3649	. . . . . . . . . . . . . 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 592	. . . . . . . . . . . . 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 736	. . . . . . . . . . . . 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 2864	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑 ∘ ^◡𝑇) ∈ (𝐴(repr‘𝑆)𝑀))
158	127	ad2antrr 736	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → Fun ^◡𝑇)
159	8, 131	eleqtrrd 2864	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑋 ∈ dom ^◡𝑇)
160	159	ad2antrr 736	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → 𝑋 ∈ dom ^◡𝑇)
161		fvco 6959	. . . . . . . . . . . . . 14 ⊢ ((Fun ^◡𝑇 ∧ 𝑋 ∈ dom ^◡𝑇) → ((𝑑 ∘ ^◡𝑇)‘𝑋) = (𝑑‘(^◡𝑇‘𝑋)))
162	158, 160, 161	syl2anc 593	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → ((𝑑 ∘ ^◡𝑇)‘𝑋) = (𝑑‘(^◡𝑇‘𝑋)))
163		f1ocnvfv 7256	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) ∧ 0 ∈ (0..^𝑆)) → ((𝑇‘0) = 𝑋 → (^◡𝑇‘𝑋) = 0))
164	163	imp 410	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) ∧ 0 ∈ (0..^𝑆)) ∧ (𝑇‘0) = 𝑋) → (^◡𝑇‘𝑋) = 0)
165	13, 11, 96, 164	syl21anc 848	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (^◡𝑇‘𝑋) = 0)
166	165	ad2antrr 736	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (^◡𝑇‘𝑋) = 0)
167	166	fveq2d 6865	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑‘(^◡𝑇‘𝑋)) = (𝑑‘0))
168		simpr 488	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → ¬ (𝑑‘0) ∈ 𝐵)
169	167, 168	eqneltrd 2881	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → ¬ (𝑑‘(^◡𝑇‘𝑋)) ∈ 𝐵)
170	162, 169	eqneltrd 2881	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → ¬ ((𝑑 ∘ ^◡𝑇)‘𝑋) ∈ 𝐵)
171		fveq1 6860	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → (𝑐‘𝑋) = ((𝑑 ∘ ^◡𝑇)‘𝑋))
172	171	eleq1d 2846	. . . . . . . . . . . . . 14 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → ((𝑐‘𝑋) ∈ 𝐵 ↔ ((𝑑 ∘ ^◡𝑇)‘𝑋) ∈ 𝐵))
173	172	notbid 320	. . . . . . . . . . . . 13 ⊢ (𝑐 = (𝑑 ∘ ^◡𝑇) → (¬ (𝑐‘𝑋) ∈ 𝐵 ↔ ¬ ((𝑑 ∘ ^◡𝑇)‘𝑋) ∈ 𝐵))
174	173	elrab 3649	. . . . . . . . . . . 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 592	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑 ∘ ^◡𝑇) ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘𝑋) ∈ 𝐵})
176	175, 18	eleqtrrdi 2872	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑑 ∈ (𝐴(repr‘𝑆)𝑀)) ∧ ¬ (𝑑‘0) ∈ 𝐵) → (𝑑 ∘ ^◡𝑇) ∈ 𝑃)
177	176	anasss 470	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → (𝑑 ∘ ^◡𝑇) ∈ 𝑃)
178		simpr 488	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) ∧ 𝑐 = (𝑑 ∘ ^◡𝑇)) → 𝑐 = (𝑑 ∘ ^◡𝑇))
179	178	coeq1d 5829	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) ∧ 𝑐 = (𝑑 ∘ ^◡𝑇)) → (𝑐 ∘ 𝑇) = ((𝑑 ∘ ^◡𝑇) ∘ 𝑇))
180	179	eqeq2d 2772	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) ∧ 𝑐 = (𝑑 ∘ ^◡𝑇)) → (𝑑 = (𝑐 ∘ 𝑇) ↔ 𝑑 = ((𝑑 ∘ ^◡𝑇) ∘ 𝑇)))
181		f1ococnv1 6830	. . . . . . . . . . . . . 14 ⊢ (𝑇:(0..^𝑆)–1-1-onto→(0..^𝑆) → (^◡𝑇 ∘ 𝑇) = ( I ↾ (0..^𝑆)))
182	13, 181	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (^◡𝑇 ∘ 𝑇) = ( I ↾ (0..^𝑆)))
183	182	adantr 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → (^◡𝑇 ∘ 𝑇) = ( I ↾ (0..^𝑆)))
184	183	coeq2d 5830	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → (𝑑 ∘ (^◡𝑇 ∘ 𝑇)) = (𝑑 ∘ ( I ↾ (0..^𝑆))))
185	114	adantrr 727	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → 𝑑:(0..^𝑆)⟶𝐴)
186		fcoi1 6732	. . . . . . . . . . . 12 ⊢ (𝑑:(0..^𝑆)⟶𝐴 → (𝑑 ∘ ( I ↾ (0..^𝑆))) = 𝑑)
187	185, 186	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → (𝑑 ∘ ( I ↾ (0..^𝑆))) = 𝑑)
188	184, 187	eqtr2d 2797	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → 𝑑 = (𝑑 ∘ (^◡𝑇 ∘ 𝑇)))
189		coass 6247	. . . . . . . . . 10 ⊢ ((𝑑 ∘ ^◡𝑇) ∘ 𝑇) = (𝑑 ∘ (^◡𝑇 ∘ 𝑇))
190	188, 189	eqtr4di 2814	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → 𝑑 = ((𝑑 ∘ ^◡𝑇) ∘ 𝑇))
191	177, 180, 190	rspcedvd 3582	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)) → ∃𝑐 ∈ 𝑃 𝑑 = (𝑐 ∘ 𝑇))
192	109, 191	impbida 810	. . . . . . 7 ⊢ (𝜑 → (∃𝑐 ∈ 𝑃 𝑑 = (𝑐 ∘ 𝑇) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)))
193	38, 192	bitrd 281	. . . . . 6 ⊢ (𝜑 → (𝑑 ∈ ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃) ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵)))
194		fveq1 6860	. . . . . . . . 9 ⊢ (𝑐 = 𝑑 → (𝑐‘0) = (𝑑‘0))
195	194	eleq1d 2846	. . . . . . . 8 ⊢ (𝑐 = 𝑑 → ((𝑐‘0) ∈ 𝐵 ↔ (𝑑‘0) ∈ 𝐵))
196	195	notbid 320	. . . . . . 7 ⊢ (𝑐 = 𝑑 → (¬ (𝑐‘0) ∈ 𝐵 ↔ ¬ (𝑑‘0) ∈ 𝐵))
197	196	elrab 3649	. . . . . 6 ⊢ (𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘0) ∈ 𝐵} ↔ (𝑑 ∈ (𝐴(repr‘𝑆)𝑀) ∧ ¬ (𝑑‘0) ∈ 𝐵))
198	193, 197	bitr4di 291	. . . . 5 ⊢ (𝜑 → (𝑑 ∈ ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃) ↔ 𝑑 ∈ {𝑐 ∈ (𝐴(repr‘𝑆)𝑀) ∣ ¬ (𝑐‘0) ∈ 𝐵}))
199	198	eqrdv 2759	. . . 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 2814	. . 3 ⊢ (𝜑 → ((𝑐 ∈ (𝐴 ↑_m (0..^𝑆)) ↦ (𝑐 ∘ 𝑇)) “ 𝑃) = 𝑂)
202	34, 35, 201	f1oeq123d 6794	. 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 234	1 ⊢ (𝜑 → 𝐹:𝑃–1-1-onto→𝑂)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ∀wral 3075 ∃wrex 3085 {crab 3413 Vcvv 3453 ⊆ wss 3902 ifcif 4477 {cpr 4581 ↦ cmpt 5178 I cid 5537 ^◡ccnv 5642 dom cdm 5643 ↾ cres 5645 “ cima 5646 ∘ ccom 5647 Fun wfun 6509 ⟶wf 6511 –1-1→wf1 6512 –1-1-onto→wf1o 6514 ‘cfv 6515 (class class class)co 7390 ↑_m cmap 8801 Fincfn 8920 0cc0 11066 ℕcn 12203 ℕ₀cn0 12474 ℤcz 12561 ..^cfzo 13652 Σcsu 15703 pmTrspcpmtr 19471 reprcrepr 34862

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7712 ax-inf2 9589 ax-cnex 11122 ax-resscn 11123 ax-1cn 11124 ax-icn 11125 ax-addcl 11126 ax-addrcl 11127 ax-mulcl 11128 ax-mulrcl 11129 ax-mulcom 11130 ax-addass 11131 ax-mulass 11132 ax-distr 11133 ax-i2m1 11134 ax-1ne0 11135 ax-1rid 11136 ax-rnegex 11137 ax-rrecex 11138 ax-cnre 11139 ax-pre-lttri 11140 ax-pre-lttrn 11141 ax-pre-ltadd 11142 ax-pre-mulgt0 11143 ax-pre-sup 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6282 df-ord 6343 df-on 6344 df-lim 6345 df-suc 6346 df-iota 6471 df-fun 6517 df-fn 6518 df-f 6519 df-f1 6520 df-fo 6521 df-f1o 6522 df-fv 6523 df-isom 6524 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7841 df-1st 7964 df-2nd 7965 df-frecs 8255 df-wrecs 8286 df-recs 8335 df-rdg 8374 df-1o 8430 df-2o 8431 df-er 8671 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-sup 9381 df-oi 9451 df-card 9890 df-pnf 11211 df-mnf 11212 df-xr 11213 df-ltxr 11214 df-le 11215 df-sub 11409 df-neg 11410 df-div 11838 df-nn 12204 df-2 12273 df-3 12274 df-n0 12475 df-z 12562 df-uz 12833 df-rp 12987 df-fz 13506 df-fzo 13653 df-seq 14008 df-exp 14068 df-hash 14337 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-clim 15505 df-sum 15704 df-pmtr 19472 df-repr 34863

This theorem is referenced by: hgt750lema 34911

W3C validator