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Mirrors > Home > MPE Home > Th. List > nrmreg | Structured version Visualization version GIF version |
Description: A normal T1 space is regular Hausdorff. In other words, a T4 space is T3 . One can get away with slightly weaker assumptions; see nrmr0reg 22941. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
nrmreg | ⊢ ((𝐽 ∈ Nrm ∧ 𝐽 ∈ Fre) → 𝐽 ∈ Reg) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | t1r0 23013 | . 2 ⊢ (𝐽 ∈ Fre → (KQ‘𝐽) ∈ Fre) | |
2 | nrmr0reg 22941 | . 2 ⊢ ((𝐽 ∈ Nrm ∧ (KQ‘𝐽) ∈ Fre) → 𝐽 ∈ Reg) | |
3 | 1, 2 | sylan2 594 | 1 ⊢ ((𝐽 ∈ Nrm ∧ 𝐽 ∈ Fre) → 𝐽 ∈ Reg) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2104 ‘cfv 6454 Frect1 22499 Regcreg 22501 Nrmcnrm 22502 KQckq 22885 |
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 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5496 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-ov 7306 df-oprab 7307 df-mpo 7308 df-1st 7859 df-2nd 7860 df-1o 8324 df-map 8644 df-topgen 17195 df-qtop 17259 df-top 22084 df-topon 22101 df-cld 22211 df-cn 22419 df-t0 22505 df-t1 22506 df-reg 22508 df-nrm 22509 df-kq 22886 df-hmeo 22947 df-hmph 22948 |
This theorem is referenced by: nrmhaus 23018 metreg 24067 |
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