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Theorem iccpnfhmeo 24894

Description: The defined bijection from [0, 1] to [0, +∞] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)

**Hypotheses**
Ref	Expression
iccpnfhmeo.f	⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))
iccpnfhmeo.k	⊢ 𝐾 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))

**Assertion**
Ref	Expression
iccpnfhmeo	⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾))

Proof of Theorem iccpnfhmeo Dummy variables 𝑦 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iccssxr 13447	. . . 4 ⊢ (0[,]1) ⊆ ℝ^*
2		xrltso 13157	. . . 4 ⊢ < Or ℝ^*
3		soss 5581	. . . 4 ⊢ ((0[,]1) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]1)))
4	1, 2, 3	mp2 9	. . 3 ⊢ < Or (0[,]1)
5		iccssxr 13447	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
6		soss 5581	. . . . 5 ⊢ ((0[,]+∞) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]+∞)))
7	5, 2, 6	mp2 9	. . . 4 ⊢ < Or (0[,]+∞)
8		sopo 5580	. . . 4 ⊢ ( < Or (0[,]+∞) → < Po (0[,]+∞))
9	7, 8	ax-mp 5	. . 3 ⊢ < Po (0[,]+∞)
10		iccpnfhmeo.f	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ (0[,]1) ↦ if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))))
11	10	iccpnfcnv 24893	. . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
12	11	simpli 483	. . . 4 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
13		f1ofo 6825	. . . 4 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) → 𝐹:(0[,]1)–onto→(0[,]+∞))
14	12, 13	ax-mp 5	. . 3 ⊢ 𝐹:(0[,]1)–onto→(0[,]+∞)
15		elicc01 13483	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
16	15	simp1bi 1145	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ∈ ℝ)
17	16	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ℝ)
18		elicc01 13483	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0[,]1) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 ≤ 1))
19	18	simp1bi 1145	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,]1) → 𝑤 ∈ ℝ)
20	19	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ ℝ)
21		1red 11236	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 1 ∈ ℝ)
22		simp3 1138	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
23	18	simp3bi 1147	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,]1) → 𝑤 ≤ 1)
24	23	3ad2ant2 1134	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ≤ 1)
25	17, 20, 21, 22, 24	ltletrd 11395	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 < 1)
26	17, 25	gtned 11370	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 1 ≠ 𝑧)
27	26	necomd 2987	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ≠ 1)
28		ifnefalse 4512	. . . . . . . 8 ⊢ (𝑧 ≠ 1 → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) = (𝑧 / (1 − 𝑧)))
29	27, 28	syl 17	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) = (𝑧 / (1 − 𝑧)))
30		breq2 5123	. . . . . . . 8 ⊢ (+∞ = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) → ((𝑧 / (1 − 𝑧)) < +∞ ↔ (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
31		breq2 5123	. . . . . . . 8 ⊢ ((𝑤 / (1 − 𝑤)) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) → ((𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤)) ↔ (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
32		1re 11235	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
33		resubcl 11547	. . . . . . . . . . . 12 ⊢ ((1 ∈ ℝ ∧ 𝑧 ∈ ℝ) → (1 − 𝑧) ∈ ℝ)
34	32, 17, 33	sylancr 587	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (1 − 𝑧) ∈ ℝ)
35		ax-1cn 11187	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
36	17	recnd 11263	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ℂ)
37		subeq0 11509	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((1 − 𝑧) = 0 ↔ 1 = 𝑧))
38	37	necon3bid 2976	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((1 − 𝑧) ≠ 0 ↔ 1 ≠ 𝑧))
39	35, 36, 38	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → ((1 − 𝑧) ≠ 0 ↔ 1 ≠ 𝑧))
40	26, 39	mpbird 257	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (1 − 𝑧) ≠ 0)
41	17, 34, 40	redivcld 12069	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) ∈ ℝ)
42	41	ltpnfd 13137	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) < +∞)
43	42	adantr 480	. . . . . . . 8 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ 𝑤 = 1) → (𝑧 / (1 − 𝑧)) < +∞)
44		simpl3 1194	. . . . . . . . . 10 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑧 < 𝑤)
45		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
46		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
47	45, 46	icopnfhmeo 24892	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) Isom < , < ((0[,)1), (0[,)+∞)) ∧ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) ∈ (((TopOpen‘ℂ_fld) ↾_t (0[,)1))Homeo((TopOpen‘ℂ_fld) ↾_t (0[,)+∞))))
48	47	simpli 483	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) Isom < , < ((0[,)1), (0[,)+∞))
49	48	a1i 11	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) Isom < , < ((0[,)1), (0[,)+∞)))
50		simp1 1136	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (0[,]1))
51		0xr 11282	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ^*
52		1xr 11294	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
53		0le1 11760	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ≤ 1
54		snunico 13496	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
55	51, 52, 53, 54	mp3an 1463	. . . . . . . . . . . . . . . . . 18 ⊢ ((0[,)1) ∪ {1}) = (0[,]1)
56	50, 55	eleqtrrdi 2845	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ((0[,)1) ∪ {1}))
57		elun 4128	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((0[,)1) ∪ {1}) ↔ (𝑧 ∈ (0[,)1) ∨ 𝑧 ∈ {1}))
58	56, 57	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 ∈ (0[,)1) ∨ 𝑧 ∈ {1}))
59	58	ord 864	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑧 ∈ (0[,)1) → 𝑧 ∈ {1}))
60		elsni 4618	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ {1} → 𝑧 = 1)
61	59, 60	syl6 35	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑧 ∈ (0[,)1) → 𝑧 = 1))
62	61	necon1ad 2949	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 ≠ 1 → 𝑧 ∈ (0[,)1)))
63	27, 62	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (0[,)1))
64	63	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑧 ∈ (0[,)1))
65		simp2 1137	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (0[,]1))
66	65, 55	eleqtrrdi 2845	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ ((0[,)1) ∪ {1}))
67		elun 4128	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ((0[,)1) ∪ {1}) ↔ (𝑤 ∈ (0[,)1) ∨ 𝑤 ∈ {1}))
68	66, 67	sylib 218	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑤 ∈ (0[,)1) ∨ 𝑤 ∈ {1}))
69	68	ord 864	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑤 ∈ (0[,)1) → 𝑤 ∈ {1}))
70		elsni 4618	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ {1} → 𝑤 = 1)
71	69, 70	syl6 35	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑤 ∈ (0[,)1) → 𝑤 = 1))
72	71	con1d 145	. . . . . . . . . . . 12 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑤 = 1 → 𝑤 ∈ (0[,)1)))
73	72	imp 406	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑤 ∈ (0[,)1))
74		isorel 7319	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) Isom < , < ((0[,)1), (0[,)+∞)) ∧ (𝑧 ∈ (0[,)1) ∧ 𝑤 ∈ (0[,)1))) → (𝑧 < 𝑤 ↔ ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) < ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤)))
75	49, 64, 73, 74	syl12anc 836	. . . . . . . . . 10 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → (𝑧 < 𝑤 ↔ ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) < ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤)))
76	44, 75	mpbid 232	. . . . . . . . 9 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) < ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤))
77		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → 𝑥 = 𝑧)
78		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (1 − 𝑥) = (1 − 𝑧))
79	77, 78	oveq12d 7423	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 / (1 − 𝑥)) = (𝑧 / (1 − 𝑧)))
80		ovex 7438	. . . . . . . . . . 11 ⊢ (𝑧 / (1 − 𝑧)) ∈ V
81	79, 45, 80	fvmpt 6986	. . . . . . . . . 10 ⊢ (𝑧 ∈ (0[,)1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) = (𝑧 / (1 − 𝑧)))
82	64, 81	syl 17	. . . . . . . . 9 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑧) = (𝑧 / (1 − 𝑧)))
83		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → 𝑥 = 𝑤)
84		oveq2 7413	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (1 − 𝑥) = (1 − 𝑤))
85	83, 84	oveq12d 7423	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑥 / (1 − 𝑥)) = (𝑤 / (1 − 𝑤)))
86		ovex 7438	. . . . . . . . . . 11 ⊢ (𝑤 / (1 − 𝑤)) ∈ V
87	85, 45, 86	fvmpt 6986	. . . . . . . . . 10 ⊢ (𝑤 ∈ (0[,)1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤) = (𝑤 / (1 − 𝑤)))
88	73, 87	syl 17	. . . . . . . . 9 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → ((𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))‘𝑤) = (𝑤 / (1 − 𝑤)))
89	76, 82, 88	3brtr3d 5150	. . . . . . . 8 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤)))
90	30, 31, 43, 89	ifbothda 4539	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
91	29, 90	eqbrtrd 5141	. . . . . 6 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
92	91	3expia 1121	. . . . 5 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) → (𝑧 < 𝑤 → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
93		eqeq1 2739	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 1 ↔ 𝑧 = 1))
94	93, 79	ifbieq2d 4527	. . . . . . 7 ⊢ (𝑥 = 𝑧 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))))
95		pnfex 11288	. . . . . . . 8 ⊢ +∞ ∈ V
96	95, 80	ifex 4551	. . . . . . 7 ⊢ if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) ∈ V
97	94, 10, 96	fvmpt 6986	. . . . . 6 ⊢ (𝑧 ∈ (0[,]1) → (𝐹‘𝑧) = if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))))
98		eqeq1 2739	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 1 ↔ 𝑤 = 1))
99	98, 85	ifbieq2d 4527	. . . . . . 7 ⊢ (𝑥 = 𝑤 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
100	95, 86	ifex 4551	. . . . . . 7 ⊢ if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) ∈ V
101	99, 10, 100	fvmpt 6986	. . . . . 6 ⊢ (𝑤 ∈ (0[,]1) → (𝐹‘𝑤) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
102	97, 101	breqan12d 5135	. . . . 5 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) → ((𝐹‘𝑧) < (𝐹‘𝑤) ↔ if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
103	92, 102	sylibrd 259	. . . 4 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) → (𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤)))
104	103	rgen2 3184	. . 3 ⊢ ∀𝑧 ∈ (0[,]1)∀𝑤 ∈ (0[,]1)(𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤))
105		soisoi 7321	. . 3 ⊢ ((( < Or (0[,]1) ∧ < Po (0[,]+∞)) ∧ (𝐹:(0[,]1)–onto→(0[,]+∞) ∧ ∀𝑧 ∈ (0[,]1)∀𝑤 ∈ (0[,]1)(𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤)))) → 𝐹 Isom < , < ((0[,]1), (0[,]+∞)))
106	4, 9, 14, 104, 105	mp4an 693	. 2 ⊢ 𝐹 Isom < , < ((0[,]1), (0[,]+∞))
107		letsr 18603	. . . . . 6 ⊢ ≤ ∈ TosetRel
108	107	elexi 3482	. . . . 5 ⊢ ≤ ∈ V
109	108	inex1 5287	. . . 4 ⊢ ( ≤ ∩ ((0[,]1) × (0[,]1))) ∈ V
110	108	inex1 5287	. . . 4 ⊢ ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) ∈ V
111		leiso 14477	. . . . . . . 8 ⊢ (((0[,]1) ⊆ ℝ^* ∧ (0[,]+∞) ⊆ ℝ^*) → (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞))))
112	1, 5, 111	mp2an 692	. . . . . . 7 ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞)))
113	106, 112	mpbi 230	. . . . . 6 ⊢ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞))
114		isores1 7327	. . . . . 6 ⊢ (𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ≤ ((0[,]1), (0[,]+∞)))
115	113, 114	mpbi 230	. . . . 5 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ≤ ((0[,]1), (0[,]+∞))
116		isores2 7326	. . . . 5 ⊢ (𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ≤ ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞)))
117	115, 116	mpbi 230	. . . 4 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞))
118		tsrps 18597	. . . . . . . 8 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
119	107, 118	ax-mp 5	. . . . . . 7 ⊢ ≤ ∈ PosetRel
120		ledm 18600	. . . . . . . 8 ⊢ ℝ^* = dom ≤
121	120	psssdm 18592	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]1) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1))
122	119, 1, 121	mp2an 692	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1)
123	122	eqcomi 2744	. . . . 5 ⊢ (0[,]1) = dom ( ≤ ∩ ((0[,]1) × (0[,]1)))
124	120	psssdm 18592	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]+∞) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞))
125	119, 5, 124	mp2an 692	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞)
126	125	eqcomi 2744	. . . . 5 ⊢ (0[,]+∞) = dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))
127	123, 126	ordthmeo 23740	. . . 4 ⊢ ((( ≤ ∩ ((0[,]1) × (0[,]1))) ∈ V ∧ ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) ∈ V ∧ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞))) → 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))))
128	109, 110, 117, 127	mp3an 1463	. . 3 ⊢ 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
129		dfii5 24829	. . . 4 ⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
130		iccpnfhmeo.k	. . . . 5 ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
131		ordtresticc 23161	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
132	130, 131	eqtri 2758	. . . 4 ⊢ 𝐾 = (ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
133	129, 132	oveq12i 7417	. . 3 ⊢ (IIHomeo𝐾) = ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
134	128, 133	eleqtrri 2833	. 2 ⊢ 𝐹 ∈ (IIHomeo𝐾)
135	106, 134	pm3.2i 470	1 ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 Vcvv 3459 ∪ cun 3924 ∩ cin 3925 ⊆ wss 3926 ifcif 4500 {csn 4601 class class class wbr 5119 ↦ cmpt 5201 Po wpo 5559 Or wor 5560 × cxp 5652 ^◡ccnv 5653 dom cdm 5654 –onto→wfo 6529 –1-1-onto→wf1o 6530 ‘cfv 6531 Isom wiso 6532 (class class class)co 7405 ℂcc 11127 ℝcr 11128 0cc0 11129 1c1 11130 + caddc 11132 +∞cpnf 11266 ℝ^*cxr 11268 < clt 11269 ≤ cle 11270 − cmin 11466 / cdiv 11894 [,)cico 13364 [,]cicc 13365 ↾_t crest 17434 TopOpenctopn 17435 ordTopcordt 17513 PosetRelcps 18574 TosetRel ctsr 18575 ℂ_fldccnfld 21315 Homeochmeo 23691 IIcii 24819

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fi 9423 df-sup 9454 df-inf 9455 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12502 df-z 12589 df-dec 12709 df-uz 12853 df-q 12965 df-rp 13009 df-xneg 13128 df-xadd 13129 df-xmul 13130 df-ioo 13366 df-ioc 13367 df-ico 13368 df-icc 13369 df-fz 13525 df-seq 14020 df-exp 14080 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-struct 17166 df-slot 17201 df-ndx 17213 df-base 17229 df-plusg 17284 df-mulr 17285 df-starv 17286 df-tset 17290 df-ple 17291 df-ds 17293 df-unif 17294 df-rest 17436 df-topn 17437 df-topgen 17457 df-ordt 17515 df-ps 18576 df-tsr 18577 df-psmet 21307 df-xmet 21308 df-met 21309 df-bl 21310 df-mopn 21311 df-cnfld 21316 df-top 22832 df-topon 22849 df-topsp 22871 df-bases 22884 df-cn 23165 df-hmeo 23693 df-xms 24259 df-ms 24260 df-ii 24821

This theorem is referenced by: xrhmeo 24895 xrge0hmph 33963

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