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Theorem metnrmlem1 24746

Description: Lemma for metnrm 24749. (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 11174	. . 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 2729	. . . . . . . . 9 ⊢ ∪ 𝐽 = ∪ 𝐽
7	6	cldss 22914	. . . . . . . 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 24327	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽)
11	3, 10	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑋 = ∪ 𝐽)
12	8, 11	sseqtrrd 3973	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑆 ⊆ 𝑋)
13		metdscn.f	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ inf(ran (𝑦 ∈ 𝑆 ↦ (𝑥𝐷𝑦)), ℝ^*, < ))
14	13	metdsf 24735	. . . . . 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 22914	. . . . . . . 8 ⊢ (𝑇 ∈ (Clsd‘𝐽) → 𝑇 ⊆ ∪ 𝐽)
19	17, 18	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ ∪ 𝐽)
20	19, 11	sseqtrrd 3973	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝑇 ⊆ 𝑋)
21		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑇)
22	20, 21	sseldd 3936	. . . . 5 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐵 ∈ 𝑋)
23	15, 22	ffvelcdmd 7019	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ (0[,]+∞))
24		eliccxr 13338	. . . 4 ⊢ ((𝐹‘𝐵) ∈ (0[,]+∞) → (𝐹‘𝐵) ∈ ℝ^*)
25	23, 24	syl 17	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ∈ ℝ^*)
26		ifcl 4522	. . 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 770	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑆)
29	12, 28	sseldd 3936	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → 𝐴 ∈ 𝑋)
30		xmetcl 24217	. . 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 13080	. . 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 24738	. . . 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 24233	. . . . . 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 24737	. . . . . 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 7367	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +_𝑒 (𝐹‘𝐴)) = ((𝐴𝐷𝐵) +_𝑒 0))
41	31	xaddridd 13145	. . . 4 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐴𝐷𝐵) +_𝑒 0) = (𝐴𝐷𝐵))
42	40, 41	eqtrd 2764	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → ((𝐵𝐷𝐴) +_𝑒 (𝐹‘𝐴)) = (𝐴𝐷𝐵))
43	35, 42	breqtrd 5118	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑇)) → (𝐹‘𝐵) ≤ (𝐴𝐷𝐵))
44	27, 25, 31, 33, 43	xrletrd 13064	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 2109 ∩ cin 3902 ⊆ wss 3903 ∅c0 4284 ifcif 4476 ∪ cuni 4858 class class class wbr 5092 ↦ cmpt 5173 ran crn 5620 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 infcinf 9331 0cc0 11009 1c1 11010 +∞cpnf 11146 ℝ^*cxr 11148 < clt 11149 ≤ cle 11150 +_𝑒 cxad 13012 [,]cicc 13251 ∞Metcxmet 21246 MetOpencmopn 21251 Clsdccld 22901

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-ec 8627 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-sup 9332 df-inf 9333 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-n0 12385 df-z 12472 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-icc 13255 df-topgen 17347 df-psmet 21253 df-xmet 21254 df-bl 21256 df-mopn 21257 df-top 22779 df-topon 22796 df-bases 22831 df-cld 22904

This theorem is referenced by: metnrmlem3 24748

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