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Theorem reparphti 25373

Description: Lemma for reparpht 25374. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.) Avoid ax-mulf 11319. (Revised by GG, 16-Mar-2025.)

**Hypotheses**
Ref	Expression
reparpht.1	⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
reparpht.2	⊢ (𝜑 → 𝐺 ∈ (II Cn II))
reparpht.3	⊢ (𝜑 → (𝐺‘0) = 0)
reparpht.4	⊢ (𝜑 → (𝐺‘1) = 1)
reparphti.5	⊢ 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 𝑢 𝑣 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reparpht.2	. . 3 ⊢ (𝜑 → 𝐺 ∈ (II Cn II))
2		reparpht.1	. . 3 ⊢ (𝜑 → 𝐹 ∈ (II Cn 𝐽))
3		cnco 23639	. . 3 ⊢ ((𝐺 ∈ (II Cn II) ∧ 𝐹 ∈ (II Cn 𝐽)) → (𝐹 ∘ 𝐺) ∈ (II Cn 𝐽))
4	1, 2, 3	syl2anc 596	. 2 ⊢ (𝜑 → (𝐹 ∘ 𝐺) ∈ (II Cn 𝐽))
5		reparphti.5	. . 3 ⊢ 𝐻 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))))
6		iitopon 25255	. . . . 5 ⊢ II ∈ (TopOn‘(0[,]1))
7	6	a1i 11	. . . 4 ⊢ (𝜑 → II ∈ (TopOn‘(0[,]1)))
8		eqid 2763	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
9	8	cnfldtop 25157	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ Top
10		cnrest2r 23660	. . . . . . . . . 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 14	. . . . . . . . 9 ⊢ (𝜑 → ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))) ⊆ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
12	7, 7	cnmpt2nd 24043	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×_t II) Cn II))
13		iirevcn 25306	. . . . . . . . . . . 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 7459	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (1 − 𝑧) = (1 − 𝑦))
16	7, 7, 12, 7, 14, 15	cnmpt21 24045	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II ×_t II) Cn II))
17	8	dfii3 25259	. . . . . . . . . . 11 ⊢ II = ((TopOpen‘ℂ_fld) ↾_t (0[,]1))
18	17	oveq2i 7462	. . . . . . . . . 10 ⊢ ((II ×_t II) Cn II) = ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1)))
19	16, 18	eleqtrdi 2874	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
20	11, 19	sseldd 3914	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (1 − 𝑦)) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
21	7, 7	cnmpt1st 24042	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×_t II) Cn II))
22	7, 7, 21, 1	cnmpt21f 24046	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×_t II) Cn II))
23	22, 18	eleqtrdi 2874	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
24	11, 23	sseldd 3914	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐺‘𝑥)) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
25	8	cnfldtopon 25156	. . . . . . . . 9 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
26	25	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ))
27	8	mpomulcn 25243	. . . . . . . . 9 ⊢ (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
28	27	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑢 ∈ ℂ, 𝑣 ∈ ℂ ↦ (𝑢 · 𝑣)) ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
29		oveq12 7460	. . . . . . . 8 ⊢ ((𝑢 = (1 − 𝑦) ∧ 𝑣 = (𝐺‘𝑥)) → (𝑢 · 𝑣) = ((1 − 𝑦) · (𝐺‘𝑥)))
30	7, 7, 20, 24, 26, 26, 28, 29	cnmpt22 24048	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((1 − 𝑦) · (𝐺‘𝑥))) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
31	9, 10	ax-mp 5	. . . . . . . . . 10 ⊢ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))) ⊆ ((II ×_t II) Cn (TopOpen‘ℂ_fld))
32	18, 31	eqsstri 3959	. . . . . . . . 9 ⊢ ((II ×_t II) Cn II) ⊆ ((II ×_t II) Cn (TopOpen‘ℂ_fld))
33	32, 12	sselid 3911	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑦) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
34	32, 21	sselid 3911	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ 𝑥) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
35		oveq12 7460	. . . . . . . 8 ⊢ ((𝑢 = 𝑦 ∧ 𝑣 = 𝑥) → (𝑢 · 𝑣) = (𝑦 · 𝑥))
36	7, 7, 33, 34, 26, 26, 28, 35	cnmpt22 24048	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝑦 · 𝑥)) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
37	8	addcn 25240	. . . . . . . 8 ⊢ + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
38	37	a1i 11	. . . . . . 7 ⊢ (𝜑 → + ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
39	7, 7, 30, 36, 38	cnmpt22f 24049	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn (TopOpen‘ℂ_fld)))
40		iiuni 25257	. . . . . . . . . . . . . . 15 ⊢ (0[,]1) = ∪ II
41	40, 40	cnf 23619	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ (II Cn II) → 𝐺:(0[,]1)⟶(0[,]1))
42	1, 41	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐺:(0[,]1)⟶(0[,]1))
43	42	ffvelcdmda 7116	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ (0[,]1)) → (𝐺‘𝑥) ∈ (0[,]1))
44	43	adantrr 730	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → (𝐺‘𝑥) ∈ (0[,]1))
45		simprl 783	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → 𝑥 ∈ (0[,]1))
46		simprr 785	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → 𝑦 ∈ (0[,]1))
47		0re 11349	. . . . . . . . . . . 12 ⊢ 0 ∈ ℝ
48		1re 11347	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
49		icccvx 25326	. . . . . . . . . . . 12 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ) → (((𝐺‘𝑥) ∈ (0[,]1) ∧ 𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1)))
50	47, 48, 49	mp2an 705	. . . . . . . . . . 11 ⊢ (((𝐺‘𝑥) ∈ (0[,]1) ∧ 𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1)) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))
51	44, 45, 46, 50	syl3anc 1399	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,]1) ∧ 𝑦 ∈ (0[,]1))) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))
52	51	ralrimivva 3209	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑥 ∈ (0[,]1)∀𝑦 ∈ (0[,]1)(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) ∈ (0[,]1))
53		eqid 2763	. . . . . . . . . 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)))
54	53	fmpo 8114	. . . . . . . . 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))
55	52, 54	sylib 221	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))):((0[,]1) × (0[,]1))⟶(0[,]1))
56	55	frnd 6748	. . . . . . 7 ⊢ (𝜑 → ran (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ⊆ (0[,]1))
57		unitsscn 13671	. . . . . . . 8 ⊢ (0[,]1) ⊆ ℂ
58	57	a1i 11	. . . . . . 7 ⊢ (𝜑 → (0[,]1) ⊆ ℂ)
59		cnrest2 23659	. . . . . . 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)))))
60	26, 56, 58, 59	syl3anc 1399	. . . . . 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)))))
61	39, 60	mpbid 235	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn ((TopOpen‘ℂ_fld) ↾_t (0[,]1))))
62	61, 18	eleqtrrdi 2875	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) ∈ ((II ×_t II) Cn II))
63	7, 7, 62, 2	cnmpt21f 24046	. . 3 ⊢ (𝜑 → (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)))) ∈ ((II ×_t II) Cn 𝐽))
64	5, 63	eqeltrid 2868	. 2 ⊢ (𝜑 → 𝐻 ∈ ((II ×_t II) Cn 𝐽))
65	42	ffvelcdmda 7116	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘𝑠) ∈ (0[,]1))
66	57, 65	sselid 3911	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘𝑠) ∈ ℂ)
67	66	mullidd 11366	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 · (𝐺‘𝑠)) = (𝐺‘𝑠))
68		elunitcn 13639	. . . . . . . 8 ⊢ (𝑠 ∈ (0[,]1) → 𝑠 ∈ ℂ)
69	68	adantl 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ ℂ)
70	69	mul02d 11547	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 · 𝑠) = 0)
71	67, 70	oveq12d 7469	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝐺‘𝑠)) + (0 · 𝑠)) = ((𝐺‘𝑠) + 0))
72	66	addridd 11549	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐺‘𝑠) + 0) = (𝐺‘𝑠))
73	71, 72	eqtrd 2798	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 · (𝐺‘𝑠)) + (0 · 𝑠)) = (𝐺‘𝑠))
74	73	fveq2d 6920	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))) = (𝐹‘(𝐺‘𝑠)))
75		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → 𝑠 ∈ (0[,]1))
76		0elunit 13640	. . . 4 ⊢ 0 ∈ (0[,]1)
77		simpr 490	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑦 = 0)
78	77	oveq2d 7467	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (1 − 𝑦) = (1 − 0))
79		1m0e1 12502	. . . . . . . . 9 ⊢ (1 − 0) = 1
80	78, 79	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (1 − 𝑦) = 1)
81		simpl 488	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → 𝑥 = 𝑠)
82	81	fveq2d 6920	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝐺‘𝑥) = (𝐺‘𝑠))
83	80, 82	oveq12d 7469	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → ((1 − 𝑦) · (𝐺‘𝑥)) = (1 · (𝐺‘𝑠)))
84	77, 81	oveq12d 7469	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝑦 · 𝑥) = (0 · 𝑠))
85	83, 84	oveq12d 7469	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = ((1 · (𝐺‘𝑠)) + (0 · 𝑠)))
86	85	fveq2d 6920	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 0) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))))
87		fvex 6929	. . . . 5 ⊢ (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))) ∈ V
88	86, 5, 87	ovmpoa 7607	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 0 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))))
89	75, 76, 88	sylancl 598	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = (𝐹‘((1 · (𝐺‘𝑠)) + (0 · 𝑠))))
90		fvco3 7016	. . . 4 ⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘𝑠) = (𝐹‘(𝐺‘𝑠)))
91	42, 90	sylan 592	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘𝑠) = (𝐹‘(𝐺‘𝑠)))
92	74, 89, 91	3eqtr4d 2808	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻0) = ((𝐹 ∘ 𝐺)‘𝑠))
93		1elunit 13641	. . . 4 ⊢ 1 ∈ (0[,]1)
94		simpr 490	. . . . . . . . . 10 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑦 = 1)
95	94	oveq2d 7467	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (1 − 𝑦) = (1 − 1))
96		1m1e0 12455	. . . . . . . . 9 ⊢ (1 − 1) = 0
97	95, 96	eqtrdi 2814	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (1 − 𝑦) = 0)
98		simpl 488	. . . . . . . . 9 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → 𝑥 = 𝑠)
99	98	fveq2d 6920	. . . . . . . 8 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝐺‘𝑥) = (𝐺‘𝑠))
100	97, 99	oveq12d 7469	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → ((1 − 𝑦) · (𝐺‘𝑥)) = (0 · (𝐺‘𝑠)))
101	94, 98	oveq12d 7469	. . . . . . 7 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝑦 · 𝑥) = (1 · 𝑠))
102	100, 101	oveq12d 7469	. . . . . 6 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = ((0 · (𝐺‘𝑠)) + (1 · 𝑠)))
103	102	fveq2d 6920	. . . . 5 ⊢ ((𝑥 = 𝑠 ∧ 𝑦 = 1) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))))
104		fvex 6929	. . . . 5 ⊢ (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))) ∈ V
105	103, 5, 104	ovmpoa 7607	. . . 4 ⊢ ((𝑠 ∈ (0[,]1) ∧ 1 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))))
106	75, 93, 105	sylancl 598	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))))
107	66	mul02d 11547	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 · (𝐺‘𝑠)) = 0)
108	69	mullidd 11366	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 · 𝑠) = 𝑠)
109	107, 108	oveq12d 7469	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝐺‘𝑠)) + (1 · 𝑠)) = (0 + 𝑠))
110	69	addlidd 11550	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0 + 𝑠) = 𝑠)
111	109, 110	eqtrd 2798	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((0 · (𝐺‘𝑠)) + (1 · 𝑠)) = 𝑠)
112	111	fveq2d 6920	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘((0 · (𝐺‘𝑠)) + (1 · 𝑠))) = (𝐹‘𝑠))
113	106, 112	eqtrd 2798	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠𝐻1) = (𝐹‘𝑠))
114		reparpht.3	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘0) = 0)
115	114	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘0) = 0)
116	115	oveq2d 7467	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘0)) = ((1 − 𝑠) · 0))
117		ax-1cn 11297	. . . . . . . . 9 ⊢ 1 ∈ ℂ
118		subcl 11595	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝑠 ∈ ℂ) → (1 − 𝑠) ∈ ℂ)
119	117, 69, 118	sylancr 599	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1 − 𝑠) ∈ ℂ)
120	119	mul01d 11548	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · 0) = 0)
121	116, 120	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘0)) = 0)
122	69	mul01d 11548	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠 · 0) = 0)
123	121, 122	oveq12d 7469	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)) = (0 + 0))
124		00id 11524	. . . . 5 ⊢ (0 + 0) = 0
125	123, 124	eqtrdi 2814	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)) = 0)
126	125	fveq2d 6920	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) = (𝐹‘0))
127		simpr 490	. . . . . . . . 9 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
128	127	oveq2d 7467	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (1 − 𝑦) = (1 − 𝑠))
129		simpl 488	. . . . . . . . 9 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → 𝑥 = 0)
130	129	fveq2d 6920	. . . . . . . 8 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝐺‘𝑥) = (𝐺‘0))
131	128, 130	oveq12d 7469	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → ((1 − 𝑦) · (𝐺‘𝑥)) = ((1 − 𝑠) · (𝐺‘0)))
132	127, 129	oveq12d 7469	. . . . . . 7 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝑦 · 𝑥) = (𝑠 · 0))
133	131, 132	oveq12d 7469	. . . . . 6 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = (((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0)))
134	133	fveq2d 6920	. . . . 5 ⊢ ((𝑥 = 0 ∧ 𝑦 = 𝑠) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))))
135		fvex 6929	. . . . 5 ⊢ (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))) ∈ V
136	134, 5, 135	ovmpoa 7607	. . . 4 ⊢ ((0 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))))
137	76, 75, 136	sylancr 599	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘0)) + (𝑠 · 0))))
138		fvco3 7016	. . . . . 6 ⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 0 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘(𝐺‘0)))
139	42, 76, 138	sylancl 598	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘(𝐺‘0)))
140	114	fveq2d 6920	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐺‘0)) = (𝐹‘0))
141	139, 140	eqtrd 2798	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘0))
142	141	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘0) = (𝐹‘0))
143	126, 137, 142	3eqtr4d 2808	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (0𝐻𝑠) = ((𝐹 ∘ 𝐺)‘0))
144		reparpht.4	. . . . . . . . 9 ⊢ (𝜑 → (𝐺‘1) = 1)
145	144	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐺‘1) = 1)
146	145	oveq2d 7467	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘1)) = ((1 − 𝑠) · 1))
147	119	mulridd 11365	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · 1) = (1 − 𝑠))
148	146, 147	eqtrd 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) · (𝐺‘1)) = (1 − 𝑠))
149	69	mulridd 11365	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝑠 · 1) = 𝑠)
150	148, 149	oveq12d 7469	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)) = ((1 − 𝑠) + 𝑠))
151		npcan 11605	. . . . . 6 ⊢ ((1 ∈ ℂ ∧ 𝑠 ∈ ℂ) → ((1 − 𝑠) + 𝑠) = 1)
152	117, 69, 151	sylancr 599	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((1 − 𝑠) + 𝑠) = 1)
153	150, 152	eqtrd 2798	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)) = 1)
154	153	fveq2d 6920	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) = (𝐹‘1))
155		simpr 490	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑦 = 𝑠)
156	155	oveq2d 7467	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (1 − 𝑦) = (1 − 𝑠))
157		simpl 488	. . . . . . . . 9 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → 𝑥 = 1)
158	157	fveq2d 6920	. . . . . . . 8 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝐺‘𝑥) = (𝐺‘1))
159	156, 158	oveq12d 7469	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → ((1 − 𝑦) · (𝐺‘𝑥)) = ((1 − 𝑠) · (𝐺‘1)))
160	155, 157	oveq12d 7469	. . . . . . 7 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝑦 · 𝑥) = (𝑠 · 1))
161	159, 160	oveq12d 7469	. . . . . 6 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥)) = (((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1)))
162	161	fveq2d 6920	. . . . 5 ⊢ ((𝑥 = 1 ∧ 𝑦 = 𝑠) → (𝐹‘(((1 − 𝑦) · (𝐺‘𝑥)) + (𝑦 · 𝑥))) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))))
163		fvex 6929	. . . . 5 ⊢ (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))) ∈ V
164	162, 5, 163	ovmpoa 7607	. . . 4 ⊢ ((1 ∈ (0[,]1) ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))))
165	93, 75, 164	sylancr 599	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = (𝐹‘(((1 − 𝑠) · (𝐺‘1)) + (𝑠 · 1))))
166		fvco3 7016	. . . . . 6 ⊢ ((𝐺:(0[,]1)⟶(0[,]1) ∧ 1 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘(𝐺‘1)))
167	42, 93, 166	sylancl 598	. . . . 5 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘(𝐺‘1)))
168	144	fveq2d 6920	. . . . 5 ⊢ (𝜑 → (𝐹‘(𝐺‘1)) = (𝐹‘1))
169	167, 168	eqtrd 2798	. . . 4 ⊢ (𝜑 → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘1))
170	169	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → ((𝐹 ∘ 𝐺)‘1) = (𝐹‘1))
171	154, 165, 170	3eqtr4d 2808	. 2 ⊢ ((𝜑 ∧ 𝑠 ∈ (0[,]1)) → (1𝐻𝑠) = ((𝐹 ∘ 𝐺)‘1))
172	4, 2, 64, 92, 113, 143, 171	isphtpy2d 25363	1 ⊢ (𝜑 → 𝐻 ∈ ((𝐹 ∘ 𝐺)(PHtpy‘𝐽)𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2147 ∀wral 3080 ⊆ wss 3900 ↦ cmpt 5192 × cxp 5656 ran crn 5659 ∘ ccom 5662 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ∈ cmpo 7453 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 + caddc 11242 · cmul 11244 − cmin 11580 [,]cicc 13519 ↾_t crest 17641 TopOpenctopn 17642 ℂ_fldccnfld 21729 Topctop 23266 TopOnctopon 23283 Cn ccn 23597 ×_t ctx 23934 IIcii 25251 PHtpycphtpy 25344

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-icc 13523 df-fz 13680 df-fzo 13829 df-seq 14185 df-exp 14245 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-mulg 19328 df-cntz 19582 df-cmn 20047 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-cnfld 21730 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cn 23600 df-cnp 23601 df-tx 23936 df-hmeo 24129 df-xms 24694 df-ms 24695 df-tms 24696 df-ii 25253 df-htpy 25346 df-phtpy 25347

This theorem is used by: reparpht 25374

W3C validator