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Mirrors > Home > MPE Home > Th. List > metnrmlem1 | Structured version Visualization version GIF version |
Description: Lemma for metnrm 23614. (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 10778 | . . 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 2738 | . . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
7 | 6 | cldss 21780 | . . . . . . . 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 23194 | . . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
11 | 3, 10 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑋 = ∪ 𝐽) |
12 | 8, 11 | sseqtrrd 3918 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ⊆ 𝑋) |
13 | metdscn.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | |
14 | 13 | metdsf 23600 | . . . . . 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 21780 | . . . . . . . 8 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽) |
19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ ∪ 𝐽) |
20 | 19, 11 | sseqtrrd 3918 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ 𝑋) |
21 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑇) | |
22 | 20, 21 | sseldd 3878 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑋) |
23 | 15, 22 | ffvelrnd 6862 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ (0[,]+∞)) |
24 | eliccxr 12909 | . . . 4 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → (𝐹‘𝐵) ∈ ℝ*) | |
25 | 23, 24 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ ℝ*) |
26 | ifcl 4459 | . . 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 771 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑆) | |
29 | 12, 28 | sseldd 3878 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑋) |
30 | xmetcl 23084 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
31 | 3, 29, 22, 30 | syl3anc 1372 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴𝐷𝐵) ∈ ℝ*) |
32 | xrmin2 12654 | . . 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 23603 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴))) |
35 | 3, 12, 22, 29, 34 | syl22anc 838 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴))) |
36 | xmetsym 23100 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵)) | |
37 | 3, 22, 29, 36 | syl3anc 1372 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵)) |
38 | 13 | metds0 23602 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) = 0) |
39 | 3, 12, 28, 38 | syl3anc 1372 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐴) = 0) |
40 | 37, 39 | oveq12d 7188 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴)) = ((𝐴𝐷𝐵) +𝑒 0)) |
41 | 31 | xaddid1d 12719 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐴𝐷𝐵) +𝑒 0) = (𝐴𝐷𝐵)) |
42 | 40, 41 | eqtrd 2773 | . . 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 12638 | 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 1542 ∈ wcel 2114 ∩ cin 3842 ⊆ wss 3843 ∅c0 4211 ifcif 4414 ∪ cuni 4796 class class class wbr 5030 ↦ cmpt 5110 ran crn 5526 ⟶wf 6335 ‘cfv 6339 (class class class)co 7170 infcinf 8978 0cc0 10615 1c1 10616 +∞cpnf 10750 ℝ*cxr 10752 < clt 10753 ≤ cle 10754 +𝑒 cxad 12588 [,]cicc 12824 ∞Metcxmet 20202 MetOpencmopn 20207 Clsdccld 21767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-ec 8322 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-n0 11977 df-z 12063 df-uz 12325 df-q 12431 df-rp 12473 df-xneg 12590 df-xadd 12591 df-xmul 12592 df-icc 12828 df-topgen 16820 df-psmet 20209 df-xmet 20210 df-bl 20212 df-mopn 20213 df-top 21645 df-topon 21662 df-bases 21697 df-cld 21770 |
This theorem is referenced by: metnrmlem3 23613 |
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