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Theorem axsegconlem7 27521

Description: Lemma for axsegcon 27525. Show that a particular ratio of distances is in the closed unit interval. (Contributed by Scott Fenton, 18-Sep-2013.)

**Hypotheses**
Ref	Expression
axsegconlem2.1	⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)
axsegconlem7.2	⊢ 𝑇 = Σ𝑝 ∈ (1...𝑁)(((𝐶‘𝑝) − (𝐷‘𝑝))↑2)

**Assertion**
Ref	Expression
axsegconlem7	⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((√‘𝑆) / ((√‘𝑆) + (√‘𝑇))) ∈ (0[,]1))

Distinct variable groups: 𝐴,𝑝 𝐵,𝑝 𝐶,𝑝 𝐷,𝑝 𝑁,𝑝 Allowed substitution hints: 𝑆(𝑝) 𝑇(𝑝)

**Proof of Theorem axsegconlem7**
Step	Hyp	Ref	Expression
1		axsegconlem7.2	. . . . 5 ⊢ 𝑇 = Σ𝑝 ∈ (1...𝑁)(((𝐶‘𝑝) − (𝐷‘𝑝))↑2)
2	1	axsegconlem5 27519	. . . 4 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → 0 ≤ (√‘𝑇))
3	2	adantl 482	. . 3 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 0 ≤ (√‘𝑇))
4		axsegconlem2.1	. . . . . 6 ⊢ 𝑆 = Σ𝑝 ∈ (1...𝑁)(((𝐴‘𝑝) − (𝐵‘𝑝))↑2)
5	4	axsegconlem4 27518	. . . . 5 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → (√‘𝑆) ∈ ℝ)
6	5	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) → (√‘𝑆) ∈ ℝ)
7	1	axsegconlem4 27518	. . . 4 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (√‘𝑇) ∈ ℝ)
8		addge01 11578	. . . 4 ⊢ (((√‘𝑆) ∈ ℝ ∧ (√‘𝑇) ∈ ℝ) → (0 ≤ (√‘𝑇) ↔ (√‘𝑆) ≤ ((√‘𝑆) + (√‘𝑇))))
9	6, 7, 8	syl2an 596	. . 3 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (0 ≤ (√‘𝑇) ↔ (√‘𝑆) ≤ ((√‘𝑆) + (√‘𝑇))))
10	3, 9	mpbid 231	. 2 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (√‘𝑆) ≤ ((√‘𝑆) + (√‘𝑇)))
11	6	adantr 481	. . 3 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (√‘𝑆) ∈ ℝ)
12	4	axsegconlem5 27519	. . . . 5 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 0 ≤ (√‘𝑆))
13	12	3adant3 1131	. . . 4 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) → 0 ≤ (√‘𝑆))
14	13	adantr 481	. . 3 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 0 ≤ (√‘𝑆))
15		readdcl 11047	. . . 4 ⊢ (((√‘𝑆) ∈ ℝ ∧ (√‘𝑇) ∈ ℝ) → ((√‘𝑆) + (√‘𝑇)) ∈ ℝ)
16	6, 7, 15	syl2an 596	. . 3 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((√‘𝑆) + (√‘𝑇)) ∈ ℝ)
17		0red 11071	. . . 4 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 0 ∈ ℝ)
18	4	axsegconlem6 27520	. . . . 5 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) → 0 < (√‘𝑆))
19	18	adantr 481	. . . 4 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 0 < (√‘𝑆))
20	17, 11, 16, 19, 10	ltletrd 11228	. . 3 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → 0 < ((√‘𝑆) + (√‘𝑇)))
21		divelunit 13319	. . 3 ⊢ ((((√‘𝑆) ∈ ℝ ∧ 0 ≤ (√‘𝑆)) ∧ (((√‘𝑆) + (√‘𝑇)) ∈ ℝ ∧ 0 < ((√‘𝑆) + (√‘𝑇)))) → (((√‘𝑆) / ((√‘𝑆) + (√‘𝑇))) ∈ (0[,]1) ↔ (√‘𝑆) ≤ ((√‘𝑆) + (√‘𝑇))))
22	11, 14, 16, 20, 21	syl22anc 836	. 2 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → (((√‘𝑆) / ((√‘𝑆) + (√‘𝑇))) ∈ (0[,]1) ↔ (√‘𝑆) ≤ ((√‘𝑆) + (√‘𝑇))))
23	10, 22	mpbird 256	1 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴 ≠ 𝐵) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((√‘𝑆) / ((√‘𝑆) + (√‘𝑇))) ∈ (0[,]1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 class class class wbr 5089 ‘cfv 6473 (class class class)co 7329 ℝcr 10963 0cc0 10964 1c1 10965 + caddc 10967 < clt 11102 ≤ cle 11103 − cmin 11298 / cdiv 11725 2c2 12121 [,]cicc 13175 ...cfz 13332 ↑cexp 13875 √csqrt 15035 Σcsu 15488 𝔼cee 27486

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-inf2 9490 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-pre-sup 11042

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-se 5570 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-isom 6482 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-1st 7891 df-2nd 7892 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-sup 9291 df-oi 9359 df-card 9788 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-div 11726 df-nn 12067 df-2 12129 df-3 12130 df-n0 12327 df-z 12413 df-uz 12676 df-rp 12824 df-ico 13178 df-icc 13179 df-fz 13333 df-fzo 13476 df-seq 13815 df-exp 13876 df-hash 14138 df-cj 14901 df-re 14902 df-im 14903 df-sqrt 15037 df-abs 15038 df-clim 15288 df-sum 15489 df-ee 27489

This theorem is referenced by: axsegcon 27525

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