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Mirrors > Home > MPE Home > Th. List > metnrmlem1 | Structured version Visualization version GIF version |
Description: Lemma for metnrm 23467. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.) |
Ref | Expression |
---|---|
metdscn.f | ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) |
metdscn.j | ⊢ 𝐽 = (MetOpen‘𝐷) |
metnrmlem.1 | ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) |
metnrmlem.2 | ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) |
metnrmlem.3 | ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) |
metnrmlem.4 | ⊢ (𝜑 → (𝑆 ∩ 𝑇) = ∅) |
Ref | Expression |
---|---|
metnrmlem1 | ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐴𝐷𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1xr 10689 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | metnrmlem.1 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | |
3 | 2 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐷 ∈ (∞Met‘𝑋)) |
4 | metnrmlem.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) | |
5 | 4 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ∈ (Clsd‘𝐽)) |
6 | eqid 2798 | . . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
7 | 6 | cldss 21634 | . . . . . . . 8 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽) |
8 | 5, 7 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ⊆ ∪ 𝐽) |
9 | metdscn.j | . . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷) | |
10 | 9 | mopnuni 23048 | . . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
11 | 3, 10 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑋 = ∪ 𝐽) |
12 | 8, 11 | sseqtrrd 3956 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ⊆ 𝑋) |
13 | metdscn.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | |
14 | 13 | metdsf 23453 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) |
15 | 3, 12, 14 | syl2anc 587 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐹:𝑋⟶(0[,]+∞)) |
16 | metnrmlem.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) | |
17 | 16 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ∈ (Clsd‘𝐽)) |
18 | 6 | cldss 21634 | . . . . . . . 8 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽) |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ ∪ 𝐽) |
20 | 19, 11 | sseqtrrd 3956 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ 𝑋) |
21 | simprr 772 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑇) | |
22 | 20, 21 | sseldd 3916 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑋) |
23 | 15, 22 | ffvelrnd 6829 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ (0[,]+∞)) |
24 | eliccxr 12813 | . . . 4 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → (𝐹‘𝐵) ∈ ℝ*) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ ℝ*) |
26 | ifcl 4469 | . . 3 ⊢ ((1 ∈ ℝ* ∧ (𝐹‘𝐵) ∈ ℝ*) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ∈ ℝ*) | |
27 | 1, 25, 26 | sylancr 590 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ∈ ℝ*) |
28 | simprl 770 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑆) | |
29 | 12, 28 | sseldd 3916 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑋) |
30 | xmetcl 22938 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
31 | 3, 29, 22, 30 | syl3anc 1368 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴𝐷𝐵) ∈ ℝ*) |
32 | xrmin2 12559 | . . 3 ⊢ ((1 ∈ ℝ* ∧ (𝐹‘𝐵) ∈ ℝ*) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐹‘𝐵)) | |
33 | 1, 25, 32 | sylancr 590 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐹‘𝐵)) |
34 | 13 | metdstri 23456 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴))) |
35 | 3, 12, 22, 29, 34 | syl22anc 837 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴))) |
36 | xmetsym 22954 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵)) | |
37 | 3, 22, 29, 36 | syl3anc 1368 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵)) |
38 | 13 | metds0 23455 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) = 0) |
39 | 3, 12, 28, 38 | syl3anc 1368 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐴) = 0) |
40 | 37, 39 | oveq12d 7153 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴)) = ((𝐴𝐷𝐵) +𝑒 0)) |
41 | 31 | xaddid1d 12624 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐴𝐷𝐵) +𝑒 0) = (𝐴𝐷𝐵)) |
42 | 40, 41 | eqtrd 2833 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴)) = (𝐴𝐷𝐵)) |
43 | 35, 42 | breqtrd 5056 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ≤ (𝐴𝐷𝐵)) |
44 | 27, 25, 31, 33, 43 | xrletrd 12543 | 1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐴𝐷𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 ifcif 4425 ∪ cuni 4800 class class class wbr 5030 ↦ cmpt 5110 ran crn 5520 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 infcinf 8889 0cc0 10526 1c1 10527 +∞cpnf 10661 ℝ*cxr 10663 < clt 10664 ≤ cle 10665 +𝑒 cxad 12493 [,]cicc 12729 ∞Metcxmet 20076 MetOpencmopn 20081 Clsdccld 21621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-ec 8274 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-n0 11886 df-z 11970 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-bases 21551 df-cld 21624 |
This theorem is referenced by: metnrmlem3 23466 |
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