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Theorem iccpnfhmeo 24117

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 13171	. . . 4 ⊢ (0[,]1) ⊆ ℝ^*
2		xrltso 12884	. . . 4 ⊢ < Or ℝ^*
3		soss 5524	. . . 4 ⊢ ((0[,]1) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]1)))
4	1, 2, 3	mp2 9	. . 3 ⊢ < Or (0[,]1)
5		iccssxr 13171	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
6		soss 5524	. . . . 5 ⊢ ((0[,]+∞) ⊆ ℝ^* → ( < Or ℝ^* → < Or (0[,]+∞)))
7	5, 2, 6	mp2 9	. . . 4 ⊢ < Or (0[,]+∞)
8		sopo 5523	. . . 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 24116	. . . . 5 ⊢ (𝐹:(0[,]1)–1-1-onto→(0[,]+∞) ∧ ^◡𝐹 = (𝑦 ∈ (0[,]+∞) ↦ if(𝑦 = +∞, 1, (𝑦 / (1 + 𝑦)))))
12	11	simpli 484	. . . 4 ⊢ 𝐹:(0[,]1)–1-1-onto→(0[,]+∞)
13		f1ofo 6732	. . . 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 13207	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ (0[,]1) ↔ (𝑧 ∈ ℝ ∧ 0 ≤ 𝑧 ∧ 𝑧 ≤ 1))
16	15	simp1bi 1144	. . . . . . . . . . 11 ⊢ (𝑧 ∈ (0[,]1) → 𝑧 ∈ ℝ)
17	16	3ad2ant1 1132	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ℝ)
18		elicc01 13207	. . . . . . . . . . . . 13 ⊢ (𝑤 ∈ (0[,]1) ↔ (𝑤 ∈ ℝ ∧ 0 ≤ 𝑤 ∧ 𝑤 ≤ 1))
19	18	simp1bi 1144	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,]1) → 𝑤 ∈ ℝ)
20	19	3ad2ant2 1133	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ ℝ)
21		1red 10985	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 1 ∈ ℝ)
22		simp3 1137	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 < 𝑤)
23	18	simp3bi 1146	. . . . . . . . . . . 12 ⊢ (𝑤 ∈ (0[,]1) → 𝑤 ≤ 1)
24	23	3ad2ant2 1133	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ≤ 1)
25	17, 20, 21, 22, 24	ltletrd 11144	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 < 1)
26	17, 25	gtned 11119	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 1 ≠ 𝑧)
27	26	necomd 3000	. . . . . . . 8 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ≠ 1)
28		ifnefalse 4472	. . . . . . . 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 5079	. . . . . . . 8 ⊢ (+∞ = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) → ((𝑧 / (1 − 𝑧)) < +∞ ↔ (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
31		breq2 5079	. . . . . . . 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 10984	. . . . . . . . . . . 12 ⊢ 1 ∈ ℝ
33		resubcl 11294	. . . . . . . . . . . 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 10938	. . . . . . . . . . . . 13 ⊢ 1 ∈ ℂ
36	17	recnd 11012	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ℂ)
37		subeq0 11256	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑧 ∈ ℂ) → ((1 − 𝑧) = 0 ↔ 1 = 𝑧))
38	37	necon3bid 2989	. . . . . . . . . . . . 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 256	. . . . . . . . . . 11 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (1 − 𝑧) ≠ 0)
41	17, 34, 40	redivcld 11812	. . . . . . . . . 10 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) ∈ ℝ)
42	41	ltpnfd 12866	. . . . . . . . 9 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) < +∞)
43	42	adantr 481	. . . . . . . 8 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ 𝑤 = 1) → (𝑧 / (1 − 𝑧)) < +∞)
44		simpl3 1192	. . . . . . . . . 10 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑧 < 𝑤)
45		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥))) = (𝑥 ∈ (0[,)1) ↦ (𝑥 / (1 − 𝑥)))
46		eqid 2739	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
47	45, 46	icopnfhmeo 24115	. . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 1135	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ (0[,]1))
51		0xr 11031	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ∈ ℝ^*
52		1xr 11043	. . . . . . . . . . . . . . . . . . 19 ⊢ 1 ∈ ℝ^*
53		0le1 11507	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ≤ 1
54		snunico 13220	. . . . . . . . . . . . . . . . . . 19 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
55	51, 52, 53, 54	mp3an 1460	. . . . . . . . . . . . . . . . . 18 ⊢ ((0[,)1) ∪ {1}) = (0[,]1)
56	50, 55	eleqtrrdi 2851	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑧 ∈ ((0[,)1) ∪ {1}))
57		elun 4084	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 ∈ ((0[,)1) ∪ {1}) ↔ (𝑧 ∈ (0[,)1) ∨ 𝑧 ∈ {1}))
58	56, 57	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 ∈ (0[,)1) ∨ 𝑧 ∈ {1}))
59	58	ord 861	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑧 ∈ (0[,)1) → 𝑧 ∈ {1}))
60		elsni 4579	. . . . . . . . . . . . . . 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 2961	. . . . . . . . . . . . 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 481	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑧 ∈ (0[,)1))
65		simp2 1136	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ (0[,]1))
66	65, 55	eleqtrrdi 2851	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → 𝑤 ∈ ((0[,)1) ∪ {1}))
67		elun 4084	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 ∈ ((0[,)1) ∪ {1}) ↔ (𝑤 ∈ (0[,)1) ∨ 𝑤 ∈ {1}))
68	66, 67	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑤 ∈ (0[,)1) ∨ 𝑤 ∈ {1}))
69	68	ord 861	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (¬ 𝑤 ∈ (0[,)1) → 𝑤 ∈ {1}))
70		elsni 4579	. . . . . . . . . . . . . 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 407	. . . . . . . . . . 11 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → 𝑤 ∈ (0[,)1))
74		isorel 7206	. . . . . . . . . . 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 834	. . . . . . . . . 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 231	. . . . . . . . 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 7292	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (1 − 𝑥) = (1 − 𝑧))
79	77, 78	oveq12d 7302	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → (𝑥 / (1 − 𝑥)) = (𝑧 / (1 − 𝑧)))
80		ovex 7317	. . . . . . . . . . 11 ⊢ (𝑧 / (1 − 𝑧)) ∈ V
81	79, 45, 80	fvmpt 6884	. . . . . . . . . 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 7292	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑤 → (1 − 𝑥) = (1 − 𝑤))
85	83, 84	oveq12d 7302	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑤 → (𝑥 / (1 − 𝑥)) = (𝑤 / (1 − 𝑤)))
86		ovex 7317	. . . . . . . . . . 11 ⊢ (𝑤 / (1 − 𝑤)) ∈ V
87	85, 45, 86	fvmpt 6884	. . . . . . . . . 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 5106	. . . . . . . 8 ⊢ (((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) ∧ ¬ 𝑤 = 1) → (𝑧 / (1 − 𝑧)) < (𝑤 / (1 − 𝑤)))
90	30, 31, 43, 89	ifbothda 4498	. . . . . . 7 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → (𝑧 / (1 − 𝑧)) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
91	29, 90	eqbrtrd 5097	. . . . . 6 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1) ∧ 𝑧 < 𝑤) → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
92	91	3expia 1120	. . . . 5 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) → (𝑧 < 𝑤 → if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) < if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤)))))
93		eqeq1 2743	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 1 ↔ 𝑧 = 1))
94	93, 79	ifbieq2d 4486	. . . . . . 7 ⊢ (𝑥 = 𝑧 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))))
95		pnfex 11037	. . . . . . . 8 ⊢ +∞ ∈ V
96	95, 80	ifex 4510	. . . . . . 7 ⊢ if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))) ∈ V
97	94, 10, 96	fvmpt 6884	. . . . . 6 ⊢ (𝑧 ∈ (0[,]1) → (𝐹‘𝑧) = if(𝑧 = 1, +∞, (𝑧 / (1 − 𝑧))))
98		eqeq1 2743	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → (𝑥 = 1 ↔ 𝑤 = 1))
99	98, 85	ifbieq2d 4486	. . . . . . 7 ⊢ (𝑥 = 𝑤 → if(𝑥 = 1, +∞, (𝑥 / (1 − 𝑥))) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
100	95, 86	ifex 4510	. . . . . . 7 ⊢ if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))) ∈ V
101	99, 10, 100	fvmpt 6884	. . . . . 6 ⊢ (𝑤 ∈ (0[,]1) → (𝐹‘𝑤) = if(𝑤 = 1, +∞, (𝑤 / (1 − 𝑤))))
102	97, 101	breqan12d 5091	. . . . 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 258	. . . 4 ⊢ ((𝑧 ∈ (0[,]1) ∧ 𝑤 ∈ (0[,]1)) → (𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤)))
104	103	rgen2 3121	. . 3 ⊢ ∀𝑧 ∈ (0[,]1)∀𝑤 ∈ (0[,]1)(𝑧 < 𝑤 → (𝐹‘𝑧) < (𝐹‘𝑤))
105		soisoi 7208	. . 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 690	. 2 ⊢ 𝐹 Isom < , < ((0[,]1), (0[,]+∞))
107		letsr 18320	. . . . . 6 ⊢ ≤ ∈ TosetRel
108	107	elexi 3452	. . . . 5 ⊢ ≤ ∈ V
109	108	inex1 5242	. . . 4 ⊢ ( ≤ ∩ ((0[,]1) × (0[,]1))) ∈ V
110	108	inex1 5242	. . . 4 ⊢ ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) ∈ V
111		leiso 14182	. . . . . . . 8 ⊢ (((0[,]1) ⊆ ℝ^* ∧ (0[,]+∞) ⊆ ℝ^*) → (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞))))
112	1, 5, 111	mp2an 689	. . . . . . 7 ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞)))
113	106, 112	mpbi 229	. . . . . 6 ⊢ 𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞))
114		isores1 7214	. . . . . 6 ⊢ (𝐹 Isom ≤ , ≤ ((0[,]1), (0[,]+∞)) ↔ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ≤ ((0[,]1), (0[,]+∞)))
115	113, 114	mpbi 229	. . . . 5 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ≤ ((0[,]1), (0[,]+∞))
116		isores2 7213	. . . . 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 229	. . . 4 ⊢ 𝐹 Isom ( ≤ ∩ ((0[,]1) × (0[,]1))), ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))((0[,]1), (0[,]+∞))
118		tsrps 18314	. . . . . . . 8 ⊢ ( ≤ ∈ TosetRel → ≤ ∈ PosetRel)
119	107, 118	ax-mp 5	. . . . . . 7 ⊢ ≤ ∈ PosetRel
120		ledm 18317	. . . . . . . 8 ⊢ ℝ^* = dom ≤
121	120	psssdm 18309	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]1) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1))
122	119, 1, 121	mp2an 689	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,]1) × (0[,]1))) = (0[,]1)
123	122	eqcomi 2748	. . . . 5 ⊢ (0[,]1) = dom ( ≤ ∩ ((0[,]1) × (0[,]1)))
124	120	psssdm 18309	. . . . . . 7 ⊢ (( ≤ ∈ PosetRel ∧ (0[,]+∞) ⊆ ℝ^*) → dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞))
125	119, 5, 124	mp2an 689	. . . . . 6 ⊢ dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞))) = (0[,]+∞)
126	125	eqcomi 2748	. . . . 5 ⊢ (0[,]+∞) = dom ( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))
127	123, 126	ordthmeo 22962	. . . 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 1460	. . 3 ⊢ 𝐹 ∈ ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
129		dfii5 24057	. . . 4 ⊢ II = (ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))
130		iccpnfhmeo.k	. . . . 5 ⊢ 𝐾 = ((ordTop‘ ≤ ) ↾_t (0[,]+∞))
131		ordtresticc 22383	. . . . 5 ⊢ ((ordTop‘ ≤ ) ↾_t (0[,]+∞)) = (ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
132	130, 131	eqtri 2767	. . . 4 ⊢ 𝐾 = (ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞))))
133	129, 132	oveq12i 7296	. . 3 ⊢ (IIHomeo𝐾) = ((ordTop‘( ≤ ∩ ((0[,]1) × (0[,]1))))Homeo(ordTop‘( ≤ ∩ ((0[,]+∞) × (0[,]+∞)))))
134	128, 133	eleqtrri 2839	. 2 ⊢ 𝐹 ∈ (IIHomeo𝐾)
135	106, 134	pm3.2i 471	1 ⊢ (𝐹 Isom < , < ((0[,]1), (0[,]+∞)) ∧ 𝐹 ∈ (IIHomeo𝐾))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 ≠ wne 2944 ∀wral 3065 Vcvv 3433 ∪ cun 3886 ∩ cin 3887 ⊆ wss 3888 ifcif 4460 {csn 4562 class class class wbr 5075 ↦ cmpt 5158 Po wpo 5502 Or wor 5503 × cxp 5588 ^◡ccnv 5589 dom cdm 5590 –onto→wfo 6435 –1-1-onto→wf1o 6436 ‘cfv 6437 Isom wiso 6438 (class class class)co 7284 ℂcc 10878 ℝcr 10879 0cc0 10880 1c1 10881 + caddc 10883 +∞cpnf 11015 ℝ^*cxr 11017 < clt 11018 ≤ cle 11019 − cmin 11214 / cdiv 11641 [,)cico 13090 [,]cicc 13091 ↾_t crest 17140 TopOpenctopn 17141 ordTopcordt 17219 PosetRelcps 18291 TosetRel ctsr 18292 ℂ_fldccnfld 20606 Homeochmeo 22913 IIcii 24047

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957 ax-pre-sup 10958

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rmo 3072 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-isom 6446 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-om 7722 df-1st 7840 df-2nd 7841 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-1o 8306 df-er 8507 df-map 8626 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-fi 9179 df-sup 9210 df-inf 9211 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-div 11642 df-nn 11983 df-2 12045 df-3 12046 df-4 12047 df-5 12048 df-6 12049 df-7 12050 df-8 12051 df-9 12052 df-n0 12243 df-z 12329 df-dec 12447 df-uz 12592 df-q 12698 df-rp 12740 df-xneg 12857 df-xadd 12858 df-xmul 12859 df-ioo 13092 df-ioc 13093 df-ico 13094 df-icc 13095 df-fz 13249 df-seq 13731 df-exp 13792 df-cj 14819 df-re 14820 df-im 14821 df-sqrt 14955 df-abs 14956 df-struct 16857 df-slot 16892 df-ndx 16904 df-base 16922 df-plusg 16984 df-mulr 16985 df-starv 16986 df-tset 16990 df-ple 16991 df-ds 16993 df-unif 16994 df-rest 17142 df-topn 17143 df-topgen 17163 df-ordt 17221 df-ps 18293 df-tsr 18294 df-psmet 20598 df-xmet 20599 df-met 20600 df-bl 20601 df-mopn 20602 df-cnfld 20607 df-top 22052 df-topon 22069 df-topsp 22091 df-bases 22105 df-cn 22387 df-hmeo 22915 df-xms 23482 df-ms 23483 df-ii 24049

This theorem is referenced by: xrhmeo 24118 xrge0hmph 31891

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