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Theorem metnrmlem1 24988

Description: Lemma for metnrm 24991. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Mario Carneiro, 4-Sep-2015.)

**Hypotheses**
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) = ∅)

**Assertion**
Ref	Expression
metnrmlem1	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐴𝐷𝐵))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐷,𝑦 𝑦,𝐽 𝑥,𝐵,𝑦 𝑥,𝑇,𝑦 𝑥,𝑆,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝜑(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐽(𝑥)

**Proof of Theorem metnrmlem1**
Step	Hyp	Ref	Expression
1		1xr 11270	. . 3 ⊢ 1 ∈ ℝ^*
2		metnrmlem.1	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋))
3	2	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐷 ∈ (∞Met‘𝑋))
4		metnrmlem.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑆 ∈ (Clsd‘𝐽))
5	4	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ∈ (Clsd‘𝐽))
6		eqid 2769	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	cldss 23157	. . . . . . . 8 ⊢ (𝑆 ∈ (Clsd‘𝐽) → 𝑆 ⊆ ∪ 𝐽)
8	5, 7	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ⊆ ∪ 𝐽)
9		metdscn.j	. . . . . . . . 9 ⊢ 𝐽 = (MetOpen‘𝐷)
10	9	mopnuni 24569	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
11	3, 10	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑋 = ∪ 𝐽)
12	8, 11	sseqtrrd 3982	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ⊆ 𝑋)
13		metdscn.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ^*, < ))
14	13	metdsf 24977	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝐹:𝑋⟶(0[,]+∞))
15	3, 12, 14	syl2anc 595	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐹:𝑋⟶(0[,]+∞))
16		metnrmlem.3	. . . . . . . . 9 ⊢ (𝜑 → 𝑇 ∈ (Clsd‘𝐽))
17	16	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ∈ (Clsd‘𝐽))
18	6	cldss 23157	. . . . . . . 8 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽)
19	17, 18	syl 18	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ ∪ 𝐽)
20	19, 11	sseqtrrd 3982	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ 𝑋)
21		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑇)
22	20, 21	sseldd 3946	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑋)
23	15, 22	ffvelcdmd 7083	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ (0[,]+∞))
24		eliccxr 13464	. . . 4 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → (𝐹‘𝐵) ∈ ℝ^*)
25	23, 24	syl 18	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ ℝ^*)
26		ifcl 4538	. . 3 ⊢ ((1 ∈ ℝ^* ∧ (𝐹‘𝐵) ∈ ℝ^) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ∈ ℝ^)
27	1, 25, 26	sylancr 598	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ∈ ℝ^*)
28		simprl 782	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑆)
29	12, 28	sseldd 3946	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑋)
30		xmetcl 24459	. . 3 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) ∈ ℝ^*)
31	3, 29, 22, 30	syl3anc 1396	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐴𝐷𝐵) ∈ ℝ^*)
32		xrmin2 13206	. . 3 ⊢ ((1 ∈ ℝ^* ∧ (𝐹‘𝐵) ∈ ℝ^*) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐹‘𝐵))
33	1, 25, 32	sylancr 598	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → if(1 ≤ (𝐹‘𝐵), 1, (𝐹‘𝐵)) ≤ (𝐹‘𝐵))
34	13	metdstri 24980	. . . 4 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋) ∧ (𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +_𝑒 (𝐹‘𝐴)))
35	3, 12, 22, 29, 34	syl22anc 851	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ≤ ((𝐵𝐷𝐴) +_𝑒 (𝐹‘𝐴)))
36		xmetsym 24475	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵))
37	3, 22, 29, 36	syl3anc 1396	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐵𝐷𝐴) = (𝐴𝐷𝐵))
38	13	metds0 24979	. . . . . 6 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑆) → (𝐹‘𝐴) = 0)
39	3, 12, 28, 38	syl3anc 1396	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐴) = 0)
40	37, 39	oveq12d 7431	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +_𝑒 (𝐹‘𝐴)) = ((𝐴𝐷𝐵) +_𝑒 0))
41	31	xaddridd 13271	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐴𝐷𝐵) +_𝑒 0) = (𝐴𝐷𝐵))
42	40, 41	eqtrd 2804	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +_𝑒 (𝐹‘𝐴)) = (𝐴𝐷𝐵))
43	35, 42	breqtrd 5141	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ≤ (𝐴𝐷𝐵))
44	27, 25, 31, 33, 43	xrletrd 13189	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 400 = wceq 1567 ∈ wcel 2149 ∩ cin 3912 ⊆ wss 3913 ∅c0 4294 ifcif 4492 ∪ cuni 4876 class class class wbr 5113 ↦ cmpt 5196 ran crn 5665 ⟶wf 6535 ‘cfv 6539 (class class class)co 7413 infcinf 9403 0cc0 11102 1c1 11103 +∞cpnf 11242 ℝ^*cxr 11244 < clt 11245 ≤ cle 11246 +_𝑒 cxad 13137 [,]cicc 13377 ∞Metcxmet 21478 MetOpencmopn 21483 Clsdccld 23144

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5261 ax-nul 5273 ax-pow 5339 ax-pr 5407 ax-un 7735 ax-cnex 11158 ax-resscn 11159 ax-1cn 11160 ax-icn 11161 ax-addcl 11162 ax-addrcl 11163 ax-mulcl 11164 ax-mulrcl 11165 ax-mulcom 11166 ax-addass 11167 ax-mulass 11168 ax-distr 11169 ax-i2m1 11170 ax-1ne0 11171 ax-1rid 11172 ax-rnegex 11173 ax-rrecex 11174 ax-cnre 11175 ax-pre-lttri 11176 ax-pre-lttrn 11177 ax-pre-ltadd 11178 ax-pre-mulgt0 11179 ax-pre-sup 11180

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4877 df-iun 4962 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6305 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6495 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-om 7865 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-er 8696 df-ec 8698 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-sup 9404 df-inf 9405 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11445 df-neg 11446 df-div 11874 df-nn 12236 df-2 12305 df-n0 12507 df-z 12594 df-uz 12865 df-q 12975 df-rp 13019 df-xneg 13139 df-xadd 13140 df-xmul 13141 df-icc 13381 df-topgen 17498 df-psmet 21485 df-xmet 21486 df-bl 21488 df-mopn 21489 df-top 23022 df-topon 23039 df-bases 23074 df-cld 23147

This theorem is referenced by: metnrmlem3 24990

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