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| Description: Lemma for metnrm 24884. (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 11320 | . . 3 ⊢ 1 ∈ ℝ* | |
| 2 | metnrmlem.1 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) | |
| 3 | 2 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐷 ∈ (∞Met‘𝑋)) | 
| 4 | metnrmlem.2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽)) | |
| 5 | 4 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ∈ (Clsd‘𝐽)) | 
| 6 | eqid 2737 | . . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 7 | 6 | cldss 23037 | . . . . . . . 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 24451 | . . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) | 
| 11 | 3, 10 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑋 = ∪ 𝐽) | 
| 12 | 8, 11 | sseqtrrd 4021 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ⊆ 𝑋) | 
| 13 | metdscn.f | . . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ*, < )) | |
| 14 | 13 | metdsf 24870 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞)) | 
| 15 | 3, 12, 14 | syl2anc 584 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐹:𝑋⟶(0[,]+∞)) | 
| 16 | metnrmlem.3 | . . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽)) | |
| 17 | 16 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ∈ (Clsd‘𝐽)) | 
| 18 | 6 | cldss 23037 | . . . . . . . 8 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽) | 
| 19 | 17, 18 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ ∪ 𝐽) | 
| 20 | 19, 11 | sseqtrrd 4021 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ 𝑋) | 
| 21 | simprr 773 | . . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑇) | |
| 22 | 20, 21 | sseldd 3984 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑋) | 
| 23 | 15, 22 | ffvelcdmd 7105 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ (0[,]+∞)) | 
| 24 | eliccxr 13475 | . . . 4 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → (𝐹‘𝐵) ∈ ℝ*) | |
| 25 | 23, 24 | syl 17 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ ℝ*) | 
| 26 | ifcl 4571 | . . 3 ⊢ ((1 ∈ ℝ* ∧ (𝐹‘𝐵) ∈ ℝ*) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ∈ ℝ*) | |
| 27 | 1, 25, 26 | sylancr 587 | . 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 3984 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑋) | 
| 30 | xmetcl 24341 | . . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ*) | |
| 31 | 3, 29, 22, 30 | syl3anc 1373 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴𝐷𝐵) ∈ ℝ*) | 
| 32 | xrmin2 13220 | . . 3 ⊢ ((1 ∈ ℝ* ∧ (𝐹‘𝐵) ∈ ℝ*) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐹‘𝐵)) | |
| 33 | 1, 25, 32 | sylancr 587 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐹‘𝐵)) | 
| 34 | 13 | metdstri 24873 | . . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴))) | 
| 35 | 3, 12, 22, 29, 34 | syl22anc 839 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴))) | 
| 36 | xmetsym 24357 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵)) | |
| 37 | 3, 22, 29, 36 | syl3anc 1373 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵)) | 
| 38 | 13 | metds0 24872 | . . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) = 0) | 
| 39 | 3, 12, 28, 38 | syl3anc 1373 | . . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐴) = 0) | 
| 40 | 37, 39 | oveq12d 7449 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴)) = ((𝐴𝐷𝐵) +𝑒 0)) | 
| 41 | 31 | xaddridd 13285 | . . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐴𝐷𝐵) +𝑒 0) = (𝐴𝐷𝐵)) | 
| 42 | 40, 41 | eqtrd 2777 | . . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +𝑒 (𝐹‘𝐴)) = (𝐴𝐷𝐵)) | 
| 43 | 35, 42 | breqtrd 5169 | . 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ≤ (𝐴𝐷𝐵)) | 
| 44 | 27, 25, 31, 33, 43 | xrletrd 13204 | 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 395 = wceq 1540 ∈ wcel 2108 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 ifcif 4525 ∪ cuni 4907 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 infcinf 9481 0cc0 11155 1c1 11156 +∞cpnf 11292 ℝ*cxr 11294 < clt 11295 ≤ cle 11296 +𝑒 cxad 13152 [,]cicc 13390 ∞Metcxmet 21349 MetOpencmopn 21354 Clsdccld 23024 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-er 8745 df-ec 8747 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-sup 9482 df-inf 9483 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-q 12991 df-rp 13035 df-xneg 13154 df-xadd 13155 df-xmul 13156 df-icc 13394 df-topgen 17488 df-psmet 21356 df-xmet 21357 df-bl 21359 df-mopn 21360 df-top 22900 df-topon 22917 df-bases 22953 df-cld 23027 | 
| This theorem is referenced by: metnrmlem3 24883 | 
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