Theorem cvmlift3 34617

Description: A general version of cvmlift 34588. 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 2742	. . . . . . . 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 1440	. . . . . . 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 3176	. . . . . 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 7377	. . . . 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 6889	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → (𝑐‘0) = (𝑓‘0))
15	14	eqeq1d 2732	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → ((𝑐‘0) = 𝑂 ↔ (𝑓‘0) = 𝑂))
16		fveq1 6889	. . . . . . . . . 10 ⊢ (𝑐 = 𝑓 → (𝑐‘1) = (𝑓‘1))
17	16	eqeq1d 2732	. . . . . . . . 9 ⊢ (𝑐 = 𝑓 → ((𝑐‘1) = 𝑎 ↔ (𝑓‘1) = 𝑎))
18		coeq2 5857	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑔 → (𝐹 ∘ 𝑑) = (𝐹 ∘ 𝑔))
19	18	eqeq1d 2732	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐)))
20		fveq1 6889	. . . . . . . . . . . . . . 15 ⊢ (𝑑 = 𝑔 → (𝑑‘0) = (𝑔‘0))
21	20	eqeq1d 2732	. . . . . . . . . . . . . 14 ⊢ (𝑑 = 𝑔 → ((𝑑‘0) = 𝑃 ↔ (𝑔‘0) = 𝑃))
22	19, 21	anbi12d 629	. . . . . . . . . . . . 13 ⊢ (𝑑 = 𝑔 → (((𝐹 ∘ 𝑑) = (𝐺 ∘ 𝑐) ∧ (𝑑‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃)))
23	22	cbvriotavw 7377	. . . . . . . . . . . 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 5857	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑓 → (𝐺 ∘ 𝑐) = (𝐺 ∘ 𝑓))
25	24	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑓 → ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ↔ (𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓)))
26	25	anbi1d 628	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑓 → (((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑐) ∧ (𝑔‘0) = 𝑃) ↔ ((𝐹 ∘ 𝑔) = (𝐺 ∘ 𝑓) ∧ (𝑔‘0) = 𝑃)))
27	26	riotabidv 7369	. . . . . . . . . . . 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 2782	. . . . . . . . . . 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 6892	. . . . . . . . . 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 2732	. . . . . . . . 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 1434	. . . . . . . 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 3233	. . . . . . 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 2742	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → ((𝑓‘1) = 𝑎 ↔ (𝑓‘1) = 𝑥))
34	33	3anbi2d 1439	. . . . . . . 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 3176	. . . . . . 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 282	. . . . . 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 7369	. . . . 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 2782	. . . 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 5260	. . 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 2730	. . . 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 34547	. . 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 34616	. 2 ⊢ (𝜑 → ∃𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
43		sconnpconn 34516	. . . 4 ⊢ (𝐾 ∈ SConn → 𝐾 ∈ PConn)
44		pconnconn 34520	. . . 4 ⊢ (𝐾 ∈ PConn → 𝐾 ∈ Conn)
45	4, 43, 44	3syl 18	. . 3 ⊢ (𝜑 → 𝐾 ∈ Conn)
46		pconnconn 34520	. . . . . 6 ⊢ (𝑥 ∈ PConn → 𝑥 ∈ Conn)
47	46	ssriv 3985	. . . . 5 ⊢ PConn ⊆ Conn
48		nllyss 23204	. . . . 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 3979	. . 3 ⊢ (𝜑 → 𝐾 ∈ 𝑛-Locally Conn)
51	1, 2, 3, 45, 50, 6, 7, 8, 9	cvmliftmo 34573	. 2 ⊢ (𝜑 → ∃*𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))
52		reu5 3376	. 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 581	1 ⊢ (𝜑 → ∃!𝑓 ∈ (𝐾 Cn 𝐶)((𝐹 ∘ 𝑓) = 𝐺 ∧ (𝑓‘𝑂) = 𝑃))

Syntax hints:   → wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068  ∃!wreu 3372  ∃*wrmo 3373  {crab 3430   ∖ cdif 3944   ∩ cin 3946   ⊆ wss 3947  ∅c0 4321  𝒫 cpw 4601  {csn 4627  ∪ cuni 4907   ↦ cmpt 5230  ^◡ccnv 5674   ↾ cres 5677   “ cima 5678   ∘ ccom 5679  ‘cfv 6542  ℩crio 7366  (class class class)co 7411  0cc0 11112  1c1 11113   ↾_t crest 17370   Cn ccn 22948  Conncconn 23135  𝑛-Locally cnlly 23189  Homeochmeo 23477  IIcii 24615  PConncpconn 34508  SConncsconn 34509   CovMap ccvm 34544

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190  ax-addf 11191  ax-mulf 11192

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7672  df-om 7858  df-1st 7977  df-2nd 7978  df-supp 8149  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-ec 8707  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-fsupp 9364  df-fi 9408  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-4 12281  df-5 12282  df-6 12283  df-7 12284  df-8 12285  df-9 12286  df-n0 12477  df-z 12563  df-dec 12682  df-uz 12827  df-q 12937  df-rp 12979  df-xneg 13096  df-xadd 13097  df-xmul 13098  df-ioo 13332  df-ico 13334  df-icc 13335  df-fz 13489  df-fzo 13632  df-fl 13761  df-seq 13971  df-exp 14032  df-hash 14295  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-clim 15436  df-sum 15637  df-struct 17084  df-sets 17101  df-slot 17119  df-ndx 17131  df-base 17149  df-ress 17178  df-plusg 17214  df-mulr 17215  df-starv 17216  df-sca 17217  df-vsca 17218  df-ip 17219  df-tset 17220  df-ple 17221  df-ds 17223  df-unif 17224  df-hom 17225  df-cco 17226  df-rest 17372  df-topn 17373  df-0g 17391  df-gsum 17392  df-topgen 17393  df-pt 17394  df-prds 17397  df-xrs 17452  df-qtop 17457  df-imas 17458  df-xps 17460  df-mre 17534  df-mrc 17535  df-acs 17537  df-mgm 18565  df-sgrp 18644  df-mnd 18660  df-submnd 18706  df-mulg 18987  df-cntz 19222  df-cmn 19691  df-psmet 21136  df-xmet 21137  df-met 21138  df-bl 21139  df-mopn 21140  df-cnfld 21145  df-top 22616  df-topon 22633  df-topsp 22655  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-cn 22951  df-cnp 22952  df-cmp 23111  df-conn 23136  df-lly 23190  df-nlly 23191  df-tx 23286  df-hmeo 23479  df-xms 24046  df-ms 24047  df-tms 24048  df-ii 24617  df-htpy 24716  df-phtpy 24717  df-phtpc 24738  df-pco 24752  df-pconn 34510  df-sconn 34511  df-cvm 34545

This theorem is referenced by: (None)