om1opn - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > om1opn		Structured version Visualization version GIF version

Theorem om1opn 23635

Description: The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)

**Hypotheses**
Ref	Expression
om1bas.o	⊢ 𝑂 = (𝐽 Ω₁ 𝑌)
om1bas.j	⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
om1bas.y	⊢ (𝜑 → 𝑌 ∈ 𝑋)
om1opn.k	⊢ 𝐾 = (TopOpen‘𝑂)
om1opn.b	⊢ (𝜑 → 𝐵 = (Base‘𝑂))

**Assertion**
Ref	Expression
om1opn	⊢ (𝜑 → 𝐾 = ((𝐽 ↑_ko II) ↾_t 𝐵))

**Proof of Theorem om1opn**
Step	Hyp	Ref	Expression
1		om1bas.o	. . . 4 ⊢ 𝑂 = (𝐽 Ω₁ 𝑌)
2		om1bas.j	. . . 4 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋))
3		om1bas.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑋)
4	1, 2, 3	om1tset 23634	. . 3 ⊢ (𝜑 → (𝐽 ↑_ko II) = (TopSet‘𝑂))
5		om1opn.b	. . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝑂))
6	4, 5	oveq12d 7167	. 2 ⊢ (𝜑 → ((𝐽 ↑_ko II) ↾_t 𝐵) = ((TopSet‘𝑂) ↾_t (Base‘𝑂)))
7		om1opn.k	. . 3 ⊢ 𝐾 = (TopOpen‘𝑂)
8		eqid 2820	. . . 4 ⊢ (Base‘𝑂) = (Base‘𝑂)
9		eqid 2820	. . . 4 ⊢ (TopSet‘𝑂) = (TopSet‘𝑂)
10	8, 9	topnval 16703	. . 3 ⊢ ((TopSet‘𝑂) ↾_t (Base‘𝑂)) = (TopOpen‘𝑂)
11	7, 10	eqtr4i 2846	. 2 ⊢ 𝐾 = ((TopSet‘𝑂) ↾_t (Base‘𝑂))
12	6, 11	syl6reqr 2874	1 ⊢ (𝜑 → 𝐾 = ((𝐽 ↑_ko II) ↾_t 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 TopSetcts 16566 ↾_t crest 16689 TopOpenctopn 16690 TopOnctopon 21513 ↑_ko cxko 22164 IIcii 23478 Ω₁ comi 23600

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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-plusg 16573 df-tset 16579 df-rest 16691 df-topn 16692 df-topon 21514 df-om1 23605

This theorem is referenced by: (None)

W3C validator