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Mirrors > Home > MPE Home > Th. List > metnrmlem1 | Structured version Visualization version GIF version |
Description: Lemma for metnrm 23472. (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 10702 | . . 3 ⊢ 1 ∈ ℝ* | |
2 | metnrmlem.1 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | |
3 | 2 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐷 ∈ (∞Met‘𝑋)) |
4 | metnrmlem.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) | |
5 | 4 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ∈ (Clsd‘𝐽)) |
6 | eqid 2823 | . . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
7 | 6 | cldss 21639 | . . . . . . . 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 23053 | . . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
11 | 3, 10 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑋 = ∪ 𝐽) |
12 | 8, 11 | sseqtrrd 4010 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ⊆ 𝑋) |
13 | metdscn.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | |
14 | 13 | metdsf 23458 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) |
15 | 3, 12, 14 | syl2anc 586 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐹:𝑋⟶(0[,]+∞)) |
16 | metnrmlem.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) | |
17 | 16 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ∈ (Clsd‘𝐽)) |
18 | 6 | cldss 21639 | . . . . . . . 8 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽) |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ ∪ 𝐽) |
20 | 19, 11 | sseqtrrd 4010 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ 𝑋) |
21 | simprr 771 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑇) | |
22 | 20, 21 | sseldd 3970 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑋) |
23 | 15, 22 | ffvelrnd 6854 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ (0[,]+∞)) |
24 | eliccxr 12826 | . . . 4 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → (𝐹‘𝐵) ∈ ℝ*) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ ℝ*) |
26 | ifcl 4513 | . . 3 ⊢ ((1 ∈ ℝ* ∧ (𝐹‘𝐵) ∈ ℝ*) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ∈ ℝ*) | |
27 | 1, 25, 26 | sylancr 589 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ∈ ℝ*) |
28 | simprl 769 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑆) | |
29 | 12, 28 | sseldd 3970 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑋) |
30 | xmetcl 22943 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
31 | 3, 29, 22, 30 | syl3anc 1367 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴𝐷𝐵) ∈ ℝ*) |
32 | xrmin2 12574 | . . 3 ⊢ ((1 ∈ ℝ* ∧ (𝐹‘𝐵) ∈ ℝ*) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐹‘𝐵)) | |
33 | 1, 25, 32 | sylancr 589 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐹‘𝐵)) |
34 | 13 | metdstri 23461 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴))) |
35 | 3, 12, 22, 29, 34 | syl22anc 836 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴))) |
36 | xmetsym 22959 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵)) | |
37 | 3, 22, 29, 36 | syl3anc 1367 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵)) |
38 | 13 | metds0 23460 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) = 0) |
39 | 3, 12, 28, 38 | syl3anc 1367 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐴) = 0) |
40 | 37, 39 | oveq12d 7176 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴)) = ((𝐴𝐷𝐵) +𝑒 0)) |
41 | 31 | xaddid1d 12639 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐴𝐷𝐵) +𝑒 0) = (𝐴𝐷𝐵)) |
42 | 40, 41 | eqtrd 2858 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴)) = (𝐴𝐷𝐵)) |
43 | 35, 42 | breqtrd 5094 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ≤ (𝐴𝐷𝐵)) |
44 | 27, 25, 31, 33, 43 | xrletrd 12558 | 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 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3937 ⊆ wss 3938 ∅c0 4293 ifcif 4469 ∪ cuni 4840 class class class wbr 5068 ↦ cmpt 5148 ran crn 5558 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 infcinf 8907 0cc0 10539 1c1 10540 +∞cpnf 10674 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 +𝑒 cxad 12508 [,]cicc 12744 ∞Metcxmet 20532 MetOpencmopn 20537 Clsdccld 21626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-ec 8293 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-inf 8909 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-topgen 16719 df-psmet 20539 df-xmet 20540 df-bl 20542 df-mopn 20543 df-top 21504 df-topon 21521 df-bases 21556 df-cld 21629 |
This theorem is referenced by: metnrmlem3 23471 |
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