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Mirrors > Home > ILE Home > Th. List > mopnex | GIF version |
Description: The topology generated by an extended metric can also be generated by a true metric. Thus, "metrizable topologies" can equivalently be defined in terms of metrics or extended metrics. (Contributed by Mario Carneiro, 26-Aug-2015.) |
Ref | Expression |
---|---|
mopnex.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
Ref | Expression |
---|---|
mopnex | ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∃𝑑 ∈ (Met‘𝑋)𝐽 = (MetOpen‘𝑑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1rp 9614 | . . 3 ⊢ 1 ∈ ℝ+ | |
2 | eqid 2170 | . . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐷𝑦), 1}, ℝ*, < )) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐷𝑦), 1}, ℝ*, < )) | |
3 | 2 | bdmet 13296 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 1 ∈ ℝ+) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐷𝑦), 1}, ℝ*, < )) ∈ (Met‘𝑋)) |
4 | 1, 3 | mpan2 423 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐷𝑦), 1}, ℝ*, < )) ∈ (Met‘𝑋)) |
5 | rpxr 9618 | . . . 4 ⊢ (1 ∈ ℝ+ → 1 ∈ ℝ*) | |
6 | 1, 5 | ax-mp 5 | . . 3 ⊢ 1 ∈ ℝ* |
7 | 0lt1 8046 | . . 3 ⊢ 0 < 1 | |
8 | mopnex.1 | . . . 4 ⊢ 𝐽 = (MetOpen‘𝐷) | |
9 | 2, 8 | bdmopn 13298 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 1 ∈ ℝ* ∧ 0 < 1) → 𝐽 = (MetOpen‘(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐷𝑦), 1}, ℝ*, < )))) |
10 | 6, 7, 9 | mp3an23 1324 | . 2 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 = (MetOpen‘(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐷𝑦), 1}, ℝ*, < )))) |
11 | fveq2 5496 | . . 3 ⊢ (𝑑 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐷𝑦), 1}, ℝ*, < )) → (MetOpen‘𝑑) = (MetOpen‘(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐷𝑦), 1}, ℝ*, < )))) | |
12 | 11 | rspceeqv 2852 | . 2 ⊢ (((𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐷𝑦), 1}, ℝ*, < )) ∈ (Met‘𝑋) ∧ 𝐽 = (MetOpen‘(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐷𝑦), 1}, ℝ*, < )))) → ∃𝑑 ∈ (Met‘𝑋)𝐽 = (MetOpen‘𝑑)) |
13 | 4, 10, 12 | syl2anc 409 | 1 ⊢ (𝐷 ∈ (∞Met‘𝑋) → ∃𝑑 ∈ (Met‘𝑋)𝐽 = (MetOpen‘𝑑)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 ∃wrex 2449 {cpr 3584 class class class wbr 3989 ‘cfv 5198 (class class class)co 5853 ∈ cmpo 5855 infcinf 6960 0cc0 7774 1c1 7775 ℝ*cxr 7953 < clt 7954 ℝ+crp 9610 ∞Metcxmet 12774 Metcmet 12775 MetOpencmopn 12779 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 ax-arch 7893 ax-caucvg 7894 |
This theorem depends on definitions: df-bi 116 df-stab 826 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-isom 5207 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-map 6628 df-sup 6961 df-inf 6962 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-3 8938 df-4 8939 df-n0 9136 df-z 9213 df-uz 9488 df-q 9579 df-rp 9611 df-xneg 9729 df-xadd 9730 df-icc 9852 df-seqfrec 10402 df-exp 10476 df-cj 10806 df-re 10807 df-im 10808 df-rsqrt 10962 df-abs 10963 df-topgen 12600 df-psmet 12781 df-xmet 12782 df-met 12783 df-bl 12784 df-mopn 12785 df-top 12790 df-bases 12835 |
This theorem is referenced by: (None) |
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