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Theorem reparphti 22797

Description: Lemma for reparpht 22798. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)

**Hypotheses**
Ref	Expression
reparpht.2	⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
reparpht.3	⊢ (𝜑 → 𝐺 ∈ (II Cn II))
reparpht.4	⊢ (𝜑 → (𝐺‘0) = 0)
reparpht.5	⊢ (𝜑 → (𝐺‘1) = 1)
reparphti.6	⊢ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))))

**Assertion**
Ref	Expression
reparphti	⊢ (𝜑 → 𝐻 ∈ ((𝐹 ∘ 𝐺)(PHtpy‘𝐽)𝐹))

Distinct variable groups: 𝑥,𝑦,𝐹 𝑥,𝐺,𝑦 𝑥,𝐽,𝑦 𝜑,𝑥,𝑦 Allowed substitution hints: 𝐻(𝑥,𝑦)

Proof of Theorem reparphti Dummy variables 𝑠 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reparpht.3	. . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn II))
2		reparpht.2	. . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
3		cnco 21070	. . 3 ⊢ ((𝐺 ∈ (II Cn II) ∧ 𝐹 ∈ (II Cn 𝐽)) → (𝐹 ∘ 𝐺) ∈ (II Cn 𝐽))
4	1, 2, 3	syl2anc 693	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (II Cn 𝐽))
5		reparphti.6	. . 3 ⊢ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))))
6		iitopon 22682	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
7	6	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
8		eqid 2622	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
9	8	cnfldtop 22587	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ Top
10		cnrest2r 21091	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ∈ Top → ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))) ⊆ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
11	9, 10	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))) ⊆ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
12	7, 7	cnmpt2nd 21472	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×_t II) Cn II))
13		iirevcn 22729	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) ↦ (1 − 𝑧)) ∈ (II Cn II)
14	13	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑧 ∈ (0[,]1) ↦ (1 − 𝑧)) ∈ (II Cn II))
15		oveq2 6658	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (1 − 𝑧) = (1 − 𝑦))
16	7, 7, 12, 7, 14, 15	cnmpt21 21474	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II ×_t II) Cn II))
17	8	dfii3 22686	. . . . . . . . . . 11 ⊢ II = ((TopOpen‘ℂ_fld) ↾_t (0[,]1))
18	17	oveq2i 6661	. . . . . . . . . 10 ⊢ ((II ×_t II) Cn II) = ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1)))
19	16, 18	syl6eleq 2711	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
20	11, 19	sseldd 3604	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
21	7, 7	cnmpt1st 21471	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×_t II) Cn II))
22	7, 7, 21, 1	cnmpt21f 21475	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×_t II) Cn II))
23	22, 18	syl6eleq 2711	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
24	11, 23	sseldd 3604	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
25	8	mulcn 22670	. . . . . . . . 9 ⊢ · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
27	7, 7, 20, 24, 26	cnmpt22f 21478	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((1 − 𝑦) · (𝐺‘𝑥))) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
28	12, 18	syl6eleq 2711	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
29	11, 28	sseldd 3604	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
30	21, 18	syl6eleq 2711	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
31	11, 30	sseldd 3604	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
32	7, 7, 29, 31, 26	cnmpt22f 21478	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑦 · 𝑥)) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
33	8	addcn 22668	. . . . . . . 8 ⊢ + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
34	33	a1i 11	. . . . . . 7 ⊢ (𝜑 → + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
35	7, 7, 27, 32, 34	cnmpt22f 21478	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
36	8	cnfldtopon 22586	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
37	36	a1i 11	. . . . . . 7 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
38		iiuni 22684	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
39	38, 38	cnf 21050	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (II Cn II) → 𝐺:(0[,]1)⟶(0[,]1))
40	1, 39	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:(0[,]1)⟶(0[,]1))
41	40	ffvelrnda 6359	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → (𝐺‘𝑥) ∈ (0[,]1))
42	41	adantrr 753	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → (𝐺‘𝑥) ∈ (0[,]1))
43		simprl 794	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → 𝑥 ∈ (0[,]1))
44		simprr 796	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → 𝑦 ∈ (0[,]1))
45		0re 10040	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
46		1re 10039	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
47		icccvx 22749	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → (((𝐺‘𝑥) ∈ (0[,]1) ∧ 𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1)))
48	45, 46, 47	mp2an 708	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑥) ∈ (0[,]1) ∧ 𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))
49	42, 43, 44, 48	syl3anc 1326	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))
50	49	ralrimivva 2971	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))
51		eqid 2622	. . . . . . . . . 10 ⊢ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)))
52	51	fmpt2 7237	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1) ↔ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))):((0[,]1) × (0[,]1))⟶(0[,]1))
53	50, 52	sylib 208	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))):((0[,]1) × (0[,]1))⟶(0[,]1))
54		frn 6053	. . . . . . . 8 ⊢ ((𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))):((0[,]1) × (0[,]1))⟶(0[,]1) → ran (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ⊆ (0[,]1))
55	53, 54	syl 17	. . . . . . 7 ⊢ (𝜑 → ran (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ⊆ (0[,]1))
56		unitssre 12319	. . . . . . . . 9 ⊢ (0[,]1) ⊆ ℝ
57		ax-resscn 9993	. . . . . . . . 9 ⊢ ℝ ⊆ ℂ
58	56, 57	sstri 3612	. . . . . . . 8 ⊢ (0[,]1) ⊆ ℂ
59	58	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0[,]1) ⊆ ℂ)
60		cnrest2 21090	. . . . . . 7 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ ran (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ⊆ (0[,]1) ∧ (0[,]1) ⊆ ℂ) → ((𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)) ↔ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1)))))
61	37, 55, 59, 60	syl3anc 1326	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)) ↔ (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1)))))
62	35, 61	mpbid 222	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
63	62, 18	syl6eleqr 2712	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn II))
64	7, 7, 63, 2	cnmpt21f 21475	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)))) ∈ ((II ×_t II) Cn 𝐽))
65	5, 64	syl5eqel 2705	. 2 ⊢ (𝜑 → 𝐻 ∈ ((II ×_t II) Cn 𝐽))
66	40	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘𝑠) ∈ (0[,]1))
67	58, 66	sseldi 3601	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘𝑠) ∈ ℂ)
68	67	mulid2d 10058	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝐺‘𝑠)) = (𝐺‘𝑠))
69	58	sseli 3599	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,]1) → 𝑠 ∈ ℂ)
70	69	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ ℂ)
71	70	mul02d 10234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 · 𝑠) = 0)
72	68, 71	oveq12d 6668	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝐺‘𝑠)) + (0 · 𝑠)) = ((𝐺‘𝑠) + 0))
73	67	addid1d 10236	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐺‘𝑠) + 0) = (𝐺‘𝑠))
74	72, 73	eqtrd 2656	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝐺‘𝑠)) + (0 · 𝑠)) = (𝐺‘𝑠))
75	74	fveq2d 6195	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))) = (𝐹‘(𝐺‘𝑠)))
76		simpr 477	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
77		0elunit 12290	. . . 4 ⊢ 0 ∈ (0[,]1)
78		simpr 477	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
79	78	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (1 − 𝑦) = (1 − 0))
80		1m0e1 11131	. . . . . . . . 9 ⊢ (1 − 0) = 1
81	79, 80	syl6eq 2672	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (1 − 𝑦) = 1)
82		simpl 473	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
83	82	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝐺‘𝑥) = (𝐺‘𝑠))
84	81, 83	oveq12d 6668	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((1 − 𝑦) · (𝐺‘𝑥)) = (1 · (𝐺‘𝑠)))
85	78, 82	oveq12d 6668	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑦 · 𝑥) = (0 · 𝑠))
86	84, 85	oveq12d 6668	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = ((1 · (𝐺‘𝑠)) + (0 · 𝑠)))
87	86	fveq2d 6195	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))))
88		fvex 6201	. . . . 5 ⊢ (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))) ∈ V
89	87, 5, 88	ovmpt2a 6791	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))))
90	76, 77, 89	sylancl 694	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))))
91		fvco3 6275	. . . 4 ⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘𝑠) = (𝐹‘(𝐺‘𝑠)))
92	40, 91	sylan 488	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘𝑠) = (𝐹‘(𝐺‘𝑠)))
93	75, 90, 92	3eqtr4d 2666	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = ((𝐹 ∘ 𝐺)‘𝑠))
94		1elunit 12291	. . . 4 ⊢ 1 ∈ (0[,]1)
95		simpr 477	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
96	95	oveq2d 6666	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (1 − 𝑦) = (1 − 1))
97		1m1e0 11089	. . . . . . . . 9 ⊢ (1 − 1) = 0
98	96, 97	syl6eq 2672	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (1 − 𝑦) = 0)
99		simpl 473	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
100	99	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝐺‘𝑥) = (𝐺‘𝑠))
101	98, 100	oveq12d 6668	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((1 − 𝑦) · (𝐺‘𝑥)) = (0 · (𝐺‘𝑠)))
102	95, 99	oveq12d 6668	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑦 · 𝑥) = (1 · 𝑠))
103	101, 102	oveq12d 6668	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = ((0 · (𝐺‘𝑠)) + (1 · 𝑠)))
104	103	fveq2d 6195	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))))
105		fvex 6201	. . . . 5 ⊢ (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))) ∈ V
106	104, 5, 105	ovmpt2a 6791	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))))
107	76, 94, 106	sylancl 694	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))))
108	67	mul02d 10234	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝐺‘𝑠)) = 0)
109	70	mulid2d 10058	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 · 𝑠) = 𝑠)
110	108, 109	oveq12d 6668	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝐺‘𝑠)) + (1 · 𝑠)) = (0 + 𝑠))
111	70	addid2d 10237	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 + 𝑠) = 𝑠)
112	110, 111	eqtrd 2656	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝐺‘𝑠)) + (1 · 𝑠)) = 𝑠)
113	112	fveq2d 6195	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))) = (𝐹‘𝑠))
114	107, 113	eqtrd 2656	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘𝑠))
115		reparpht.4	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘0) = 0)
116	115	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘0) = 0)
117	116	oveq2d 6666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘0)) = ((1 − 𝑠) · 0))
118		ax-1cn 9994	. . . . . . . . 9 ⊢ 1 ∈ ℂ
119		subcl 10280	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝑠 ∈ ℂ) → (1 − 𝑠) ∈ ℂ)
120	118, 70, 119	sylancr 695	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 − 𝑠) ∈ ℂ)
121	120	mul01d 10235	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · 0) = 0)
122	117, 121	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘0)) = 0)
123	70	mul01d 10235	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠 · 0) = 0)
124	122, 123	oveq12d 6668	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)) = (0 + 0))
125		00id 10211	. . . . 5 ⊢ (0 + 0) = 0
126	124, 125	syl6eq 2672	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)) = 0)
127	126	fveq2d 6195	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) = (𝐹‘0))
128		simpr 477	. . . . . . . . 9 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
129	128	oveq2d 6666	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (1 − 𝑦) = (1 − 𝑠))
130		simpl 473	. . . . . . . . 9 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 = 0)
131	130	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝐺‘𝑥) = (𝐺‘0))
132	129, 131	oveq12d 6668	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ((1 − 𝑦) · (𝐺‘𝑥)) = ((1 − 𝑠) · (𝐺‘0)))
133	128, 130	oveq12d 6668	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑦 · 𝑥) = (𝑠 · 0))
134	132, 133	oveq12d 6668	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)))
135	134	fveq2d 6195	. . . . 5 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))))
136		fvex 6201	. . . . 5 ⊢ (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) ∈ V
137	135, 5, 136	ovmpt2a 6791	. . . 4 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))))
138	77, 76, 137	sylancr 695	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))))
139		fvco3 6275	. . . . . 6 ⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘(𝐺‘0)))
140	40, 77, 139	sylancl 694	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘(𝐺‘0)))
141	115	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐺‘0)) = (𝐹‘0))
142	140, 141	eqtrd 2656	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘0))
143	142	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘0))
144	127, 138, 143	3eqtr4d 2666	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = ((𝐹 ∘ 𝐺)‘0))
145		reparpht.5	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘1) = 1)
146	145	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘1) = 1)
147	146	oveq2d 6666	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘1)) = ((1 − 𝑠) · 1))
148	120	mulid1d 10057	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · 1) = (1 − 𝑠))
149	147, 148	eqtrd 2656	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘1)) = (1 − 𝑠))
150	70	mulid1d 10057	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠 · 1) = 𝑠)
151	149, 150	oveq12d 6668	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)) = ((1 − 𝑠) + 𝑠))
152		npcan 10290	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝑠 ∈ ℂ) → ((1 − 𝑠) + 𝑠) = 1)
153	118, 70, 152	sylancr 695	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) + 𝑠) = 1)
154	151, 153	eqtrd 2656	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)) = 1)
155	154	fveq2d 6195	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) = (𝐹‘1))
156		simpr 477	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
157	156	oveq2d 6666	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (1 − 𝑦) = (1 − 𝑠))
158		simpl 473	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑥 = 1)
159	158	fveq2d 6195	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝐺‘𝑥) = (𝐺‘1))
160	157, 159	oveq12d 6668	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((1 − 𝑦) · (𝐺‘𝑥)) = ((1 − 𝑠) · (𝐺‘1)))
161	156, 158	oveq12d 6668	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑦 · 𝑥) = (𝑠 · 1))
162	160, 161	oveq12d 6668	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)))
163	162	fveq2d 6195	. . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))))
164		fvex 6201	. . . . 5 ⊢ (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) ∈ V
165	163, 5, 164	ovmpt2a 6791	. . . 4 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))))
166	94, 76, 165	sylancr 695	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))))
167		fvco3 6275	. . . . . 6 ⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘(𝐺‘1)))
168	40, 94, 167	sylancl 694	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘(𝐺‘1)))
169	145	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐺‘1)) = (𝐹‘1))
170	168, 169	eqtrd 2656	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘1))
171	170	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘1))
172	155, 166, 171	3eqtr4d 2666	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = ((𝐹 ∘ 𝐺)‘1))
173	4, 2, 65, 93, 114, 144, 172	isphtpy2d 22786	1 ⊢ (𝜑 → 𝐻 ∈ ((𝐹 ∘ 𝐺)(PHtpy‘𝐽)𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ↦ cmpt 4729 × cxp 5112 ran crn 5115 ∘ ccom 5118 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↦ cmpt2 6652 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 [,]cicc 12178 ↾_t crest 16081 TopOpenctopn 16082 ℂ_fldccnfld 19746 Topctop 20698 TopOnctopon 20715 Cn ccn 21028 ×_t ctx 21363 IIcii 22678 PHtpycphtpy 22767

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cn 21031 df-cnp 21032 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-ii 22680 df-htpy 22769 df-phtpy 22770

This theorem is referenced by: reparpht 22798

W3C validator