Theorem cvmlift3 35333

Description: A general version of cvmlift 35304. 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 2749	. . . . . . . 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 3179	. . . . . 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 7398	. . . . 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 6905	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → (𝑐‘0) = (𝑓‘0))
15	14	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → ((𝑐‘0) = 𝑂 ↔ (𝑓‘0) = 𝑂))
16		fveq1 6905	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → (𝑐‘1) = (𝑓‘1))
17	16	eqeq1d 2739	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → ((𝑐‘1) = 𝑎 ↔ (𝑓‘1) = 𝑎))
18		coeq2 5869	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑔 → (𝐹 ∘ 𝑑) = (𝐹 ∘ 𝑔))
19	18	eqeq1d 2739	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐)))
20		fveq1 6905	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑔 → (𝑑‘0) = (𝑔‘0))
21	20	eqeq1d 2739	. . . . . . . . . . . . . 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 7398	. . . . . . . . . . . 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 5869	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑓 → (𝐺 ∘ 𝑐) = (𝐺 ∘ 𝑓))
25	24	eqeq2d 2748	. . . . . . . . . . . . . 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 7390	. . . . . . . . . . . 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 2789	. . . . . . . . . . 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 6908	. . . . . . . . . 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 2739	. . . . . . . . 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 3238	. . . . . . 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 2749	. . . . . . . . 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 3179	. . . . . . 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 7390	. . . . 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 2789	. . . 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 5255	. . 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 2737	. . . 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 35263	. . 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 35332	. 2 ⊢ (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
43		sconnpconn 35232	. . . 4 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ PConn)
44		pconnconn 35236	. . . 4 ⊢ (𝐾 ∈ PConn → 𝐾 ∈ Conn)
45	4, 43, 44	3syl 18	. . 3 ⊢ (𝜑 → 𝐾 ∈ Conn)
46		pconnconn 35236	. . . . . 6 ⊢ (𝑥 ∈ PConn → 𝑥 ∈ Conn)
47	46	ssriv 3987	. . . . 5 ⊢ PConn ⊆ Conn
48		nllyss 23488	. . . . 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 3981	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
51	1, 2, 3, 45, 50, 6, 7, 8, 9	cvmliftmo 35289	. 2 ⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
52		reu5 3382	. 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 1087   = wceq 1540   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  ∃!wreu 3378  ∃*wrmo 3379  {crab 3436   ∖ cdif 3948   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {csn 4626  ∪ cuni 4907   ↦ cmpt 5225  ^◡ccnv 5684   ↾ cres 5687   “ cima 5688   ∘ ccom 5689  ‘cfv 6561  ℩crio 7387  (class class class)co 7431  0cc0 11155  1c1 11156   ↾_t crest 17465   Cn ccn 23232  Conncconn 23419  𝑛-Locally cnlly 23473  Homeochmeo 23761  IIcii 24901  PConncpconn 35224  SConncsconn 35225   CovMap ccvm 35260

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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233  ax-addf 11234

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-supp 8186  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-ec 8747  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fsupp 9402  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-sum 15723  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-mulr 17311  df-starv 17312  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-unif 17320  df-hom 17321  df-cco 17322  df-rest 17467  df-topn 17468  df-0g 17486  df-gsum 17487  df-topgen 17488  df-pt 17489  df-prds 17492  df-xrs 17547  df-qtop 17552  df-imas 17553  df-xps 17555  df-mre 17629  df-mrc 17630  df-acs 17632  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-mulg 19086  df-cntz 19335  df-cmn 19800  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-cnfld 21365  df-top 22900  df-topon 22917  df-topsp 22939  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-cn 23235  df-cnp 23236  df-cmp 23395  df-conn 23420  df-lly 23474  df-nlly 23475  df-tx 23570  df-hmeo 23763  df-xms 24330  df-ms 24331  df-tms 24332  df-ii 24903  df-cncf 24904  df-htpy 25002  df-phtpy 25003  df-phtpc 25024  df-pco 25038  df-pconn 35226  df-sconn 35227  df-cvm 35261

This theorem is referenced by: (None)