Theorem cvmlift3 35393

Description: A general version of cvmlift 35364. If 𝐾 is simply connected and weakly locally path-connected, then there is a unique lift of functions on 𝐾 which commutes with the covering map. (Contributed by Mario Carneiro, 9-Jul-2015.)

Hypotheses

Ref	Expression
cvmlift3.b	⊢ 𝐵 = ∪ 𝐶
cvmlift3.y	⊢ 𝑌 = ∪ 𝐾
cvmlift3.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
cvmlift3.k	⊢ (𝜑 → 𝐾 ∈ SConn)
cvmlift3.l	⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
cvmlift3.o	⊢ (𝜑 → 𝑂 ∈ 𝑌)
cvmlift3.g	⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
cvmlift3.p	⊢ (𝜑 → 𝑃 ∈ 𝐵)
cvmlift3.e	⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))

Assertion

Ref	Expression
cvmlift3	⊢ (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Distinct variable groups:   𝑓,𝐽   𝑓,𝐹   𝐵,𝑓   𝑓,𝐺   𝐶,𝑓   𝜑,𝑓   𝑓,𝐾   𝑃,𝑓   𝑓,𝑂   𝑓,𝑌

Proof of Theorem cvmlift3

Dummy variables 𝑏 𝑐 𝑑 𝑘 𝑠 𝑧 𝑔 𝑎 𝑢 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cvmlift3.b	. . 3 ⊢ 𝐵 = ∪ 𝐶
2		cvmlift3.y	. . 3 ⊢ 𝑌 = ∪ 𝐾
3		cvmlift3.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽))
4		cvmlift3.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ SConn)
5		cvmlift3.l	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally PConn)
6		cvmlift3.o	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑌)
7		cvmlift3.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐾 Cn 𝐽))
8		cvmlift3.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐵)
9		cvmlift3.e	. . 3 ⊢ (𝜑 → (𝐹‘𝑃) = (𝐺‘𝑂))
10		eqeq2 2745	. . . . . . . 8 ⊢ (𝑏 = 𝑧 → (((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏 ↔ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧))
11	10	3anbi3d 1444	. . . . . . 7 ⊢ (𝑏 = 𝑧 → (((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏) ↔ ((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧)))
12	11	rexbidv 3157	. . . . . 6 ⊢ (𝑏 = 𝑧 → (∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏) ↔ ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧)))
13	12	cbvriotavw 7319	. . . . 5 ⊢ (℩𝑏 ∈ 𝐵 ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏)) = (℩𝑧 ∈ 𝐵 ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧))
14		fveq1 6827	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → (𝑐‘0) = (𝑓‘0))
15	14	eqeq1d 2735	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → ((𝑐‘0) = 𝑂 ↔ (𝑓‘0) = 𝑂))
16		fveq1 6827	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → (𝑐‘1) = (𝑓‘1))
17	16	eqeq1d 2735	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → ((𝑐‘1) = 𝑎 ↔ (𝑓‘1) = 𝑎))
18		coeq2 5802	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑔 → (𝐹 ∘ 𝑑) = (𝐹 ∘ 𝑔))
19	18	eqeq1d 2735	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐)))
20		fveq1 6827	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑔 → (𝑑‘0) = (𝑔‘0))
21	20	eqeq1d 2735	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((𝑑‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
22	19, 21	anbi12d 632	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑔 → (((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃)))
23	22	cbvriotavw 7319	. . . . . . . . . . . 12 ⊢ (℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃))
24		coeq2 5802	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑓 → (𝐺 ∘ 𝑐) = (𝐺 ∘ 𝑓))
25	24	eqeq2d 2744	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑓 → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓)))
26	25	anbi1d 631	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑓 → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))
27	26	riotabidv 7311	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑓 → (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))
28	23, 27	eqtrid 2780	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑓 → (℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃)) = (℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))
29	28	fveq1d 6830	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1))
30	29	eqeq1d 2735	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → (((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧 ↔ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
31	15, 17, 30	3anbi123d 1438	. . . . . . . 8 ⊢ (𝑐 = 𝑓 → (((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧) ↔ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
32	31	cbvrexvw 3212	. . . . . . 7 ⊢ (∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧))
33		eqeq2 2745	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → ((𝑓‘1) = 𝑎 ↔ (𝑓‘1) = 𝑥))
34	33	3anbi2d 1443	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → (((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
35	34	rexbidv 3157	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑎 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
36	32, 35	bitrid 283	. . . . . 6 ⊢ (𝑎 = 𝑥 → (∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧) ↔ ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
37	36	riotabidv 7311	. . . . 5 ⊢ (𝑎 = 𝑥 → (℩𝑧 ∈ 𝐵 ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑧)) = (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
38	13, 37	eqtrid 2780	. . . 4 ⊢ (𝑎 = 𝑥 → (℩𝑏 ∈ 𝐵 ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏)) = (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
39	38	cbvmptv 5197	. . 3 ⊢ (𝑎 ∈ 𝑌 ↦ (℩𝑏 ∈ 𝐵 ∃𝑐 ∈ (II Cn 𝐾)((𝑐‘0) = 𝑂 ∧ (𝑐‘1) = 𝑎 ∧ ((℩𝑑 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃))‘1) = 𝑏))) = (𝑥 ∈ 𝑌 ↦ (℩𝑧 ∈ 𝐵 ∃𝑓 ∈ (II Cn 𝐾)((𝑓‘0) = 𝑂 ∧ (𝑓‘1) = 𝑥 ∧ ((℩𝑔 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃))‘1) = 𝑧)))
40		eqid 2733	. . . 4 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))}) = (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))})
41	40	cvmscbv 35323	. . 3 ⊢ (𝑘 ∈ 𝐽 ↦ {𝑠 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑠 = (◡𝐹 “ 𝑘) ∧ ∀𝑐 ∈ 𝑠 (∀𝑑 ∈ (𝑠 ∖ {𝑐})(𝑐 ∩ 𝑑) = ∅ ∧ (𝐹 ↾ 𝑐) ∈ ((𝐶 ↾t 𝑐)Homeo(𝐽 ↾t 𝑘))))}) = (𝑎 ∈ 𝐽 ↦ {𝑏 ∈ (𝒫 𝐶 ∖ {∅}) ∣ (∪ 𝑏 = (◡𝐹 “ 𝑎) ∧ ∀𝑣 ∈ 𝑏 (∀𝑢 ∈ (𝑏 ∖ {𝑣})(𝑣 ∩ 𝑢) = ∅ ∧ (𝐹 ↾ 𝑣) ∈ ((𝐶 ↾t 𝑣)Homeo(𝐽 ↾t 𝑎))))})
42	1, 2, 3, 4, 5, 6, 7, 8, 9, 39, 41	cvmlift3lem9 35392	. 2 ⊢ (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
43		sconnpconn 35292	. . . 4 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ PConn)
44		pconnconn 35296	. . . 4 ⊢ (𝐾 ∈ PConn → 𝐾 ∈ Conn)
45	4, 43, 44	3syl 18	. . 3 ⊢ (𝜑 → 𝐾 ∈ Conn)
46		pconnconn 35296	. . . . . 6 ⊢ (𝑥 ∈ PConn → 𝑥 ∈ Conn)
47	46	ssriv 3934	. . . . 5 ⊢ PConn ⊆ Conn
48		nllyss 23396	. . . . 5 ⊢ (PConn ⊆ Conn → 𝑛-Locally PConn ⊆ 𝑛-Locally Conn)
49	47, 48	ax-mp 5	. . . 4 ⊢ 𝑛-Locally PConn ⊆ 𝑛-Locally Conn
50	49, 5	sselid 3928	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
51	1, 2, 3, 45, 50, 6, 7, 8, 9	cvmliftmo 35349	. 2 ⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
52		reu5 3349	. 2 ⊢ (∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ↔ (∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃) ∧ ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃)))
53	42, 51, 52	sylanbrc 583	1 ⊢ (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  ∃!wreu 3345  ∃*wrmo 3346  {crab 3396   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  {csn 4575  ∪ cuni 4858   ↦ cmpt 5174  ^◡ccnv 5618   ↾ cres 5621   “ cima 5622   ∘ ccom 5623  ‘cfv 6486  ℩crio 7308  (class class class)co 7352  0cc0 11013  1c1 11014   ↾_t crest 17326   Cn ccn 23140  Conncconn 23327  𝑛-Locally cnlly 23381  Homeochmeo 23669  IIcii 24796  PConncpconn 35284  SConncsconn 35285   CovMap ccvm 35320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091  ax-addf 11092

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-tp 4580  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-of 7616  df-om 7803  df-1st 7927  df-2nd 7928  df-supp 8097  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-er 8628  df-ec 8630  df-map 8758  df-ixp 8828  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-fsupp 9253  df-fi 9302  df-sup 9333  df-inf 9334  df-oi 9403  df-card 9839  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-4 12197  df-5 12198  df-6 12199  df-7 12200  df-8 12201  df-9 12202  df-n0 12389  df-z 12476  df-dec 12595  df-uz 12739  df-q 12849  df-rp 12893  df-xneg 13013  df-xadd 13014  df-xmul 13015  df-ioo 13251  df-ico 13253  df-icc 13254  df-fz 13410  df-fzo 13557  df-fl 13698  df-seq 13911  df-exp 13971  df-hash 14240  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-clim 15397  df-sum 15596  df-struct 17060  df-sets 17077  df-slot 17095  df-ndx 17107  df-base 17123  df-ress 17144  df-plusg 17176  df-mulr 17177  df-starv 17178  df-sca 17179  df-vsca 17180  df-ip 17181  df-tset 17182  df-ple 17183  df-ds 17185  df-unif 17186  df-hom 17187  df-cco 17188  df-rest 17328  df-topn 17329  df-0g 17347  df-gsum 17348  df-topgen 17349  df-pt 17350  df-prds 17353  df-xrs 17408  df-qtop 17413  df-imas 17414  df-xps 17416  df-mre 17490  df-mrc 17491  df-acs 17493  df-mgm 18550  df-sgrp 18629  df-mnd 18645  df-submnd 18694  df-mulg 18983  df-cntz 19231  df-cmn 19696  df-psmet 21285  df-xmet 21286  df-met 21287  df-bl 21288  df-mopn 21289  df-cnfld 21294  df-top 22810  df-topon 22827  df-topsp 22849  df-bases 22862  df-cld 22935  df-ntr 22936  df-cls 22937  df-nei 23014  df-cn 23143  df-cnp 23144  df-cmp 23303  df-conn 23328  df-lly 23382  df-nlly 23383  df-tx 23478  df-hmeo 23671  df-xms 24236  df-ms 24237  df-tms 24238  df-ii 24798  df-cncf 24799  df-htpy 24897  df-phtpy 24898  df-phtpc 24919  df-pco 24933  df-pconn 35286  df-sconn 35287  df-cvm 35321

This theorem is referenced by: (None)