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Theorem xrsdsreclb 20896

Description: The metric of the extended real number structure is only real when both arguments are real. (Contributed by Mario Carneiro, 3-Sep-2015.)

**Hypothesis**
Ref	Expression
xrsds.d	⊢ 𝐷 = (dist‘ℝ^*_𝑠)

**Assertion**
Ref	Expression
xrsdsreclb	⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) → ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))

**Proof of Theorem xrsdsreclb**
Step	Hyp	Ref	Expression
1		xrsds.d	. . . . . 6 ⊢ 𝐷 = (dist‘ℝ^*_𝑠)
2	1	xrsdsval 20893	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴𝐷𝐵) = if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)))
3	2	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) → (𝐴𝐷𝐵) = if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)))
4	3	eleq1d 2817	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) → ((𝐴𝐷𝐵) ∈ ℝ ↔ if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)) ∈ ℝ))
5		eleq1 2820	. . . . 5 ⊢ ((𝐵 +_𝑒 -_𝑒𝐴) = if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)) → ((𝐵 +_𝑒 -_𝑒𝐴) ∈ ℝ ↔ if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)) ∈ ℝ))
6	5	imbi1d 341	. . . 4 ⊢ ((𝐵 +_𝑒 -_𝑒𝐴) = if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)) → (((𝐵 +_𝑒 -_𝑒𝐴) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) ↔ (if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))))
7		eleq1 2820	. . . . 5 ⊢ ((𝐴 +_𝑒 -_𝑒𝐵) = if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)) → ((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ ↔ if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)) ∈ ℝ))
8	7	imbi1d 341	. . . 4 ⊢ ((𝐴 +_𝑒 -_𝑒𝐵) = if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)) → (((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)) ↔ (if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))))
9	1	xrsdsreclblem 20895	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) ∧ 𝐴 ≤ 𝐵) → ((𝐵 +_𝑒 -_𝑒𝐴) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))
10		xrletri 13097	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
11	10	3adant3 1132	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) → (𝐴 ≤ 𝐵 ∨ 𝐵 ≤ 𝐴))
12	11	orcanai 1001	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) ∧ ¬ 𝐴 ≤ 𝐵) → 𝐵 ≤ 𝐴)
13		necom 2993	. . . . . . . . 9 ⊢ (𝐴 ≠ 𝐵 ↔ 𝐵 ≠ 𝐴)
14	13	3anbi3i 1159	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) ↔ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≠ 𝐴))
15		3ancoma 1098	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐵 ≠ 𝐴) ↔ (𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ≠ 𝐴))
16	14, 15	bitri 274	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) ↔ (𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ≠ 𝐴))
17	1	xrsdsreclblem 20895	. . . . . . 7 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐵 ≠ 𝐴) ∧ 𝐵 ≤ 𝐴) → ((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ → (𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ)))
18	16, 17	sylanb 581	. . . . . 6 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) ∧ 𝐵 ≤ 𝐴) → ((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ → (𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ)))
19		ancom 461	. . . . . 6 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
20	18, 19	syl6ib 250	. . . . 5 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) ∧ 𝐵 ≤ 𝐴) → ((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))
21	12, 20	syldan 591	. . . 4 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) ∧ ¬ 𝐴 ≤ 𝐵) → ((𝐴 +_𝑒 -_𝑒𝐵) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))
22	6, 8, 9, 21	ifbothda 4544	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) → (if(𝐴 ≤ 𝐵, (𝐵 +_𝑒 -_𝑒𝐴), (𝐴 +_𝑒 -_𝑒𝐵)) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))
23	4, 22	sylbid 239	. 2 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) → ((𝐴𝐷𝐵) ∈ ℝ → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))
24	1	xrsdsreval 20894	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵)))
25		recn 11165	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
26		recn 11165	. . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ)
27		subcl 11424	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ)
28	25, 26, 27	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) ∈ ℂ)
29	28	abscld 15348	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (abs‘(𝐴 − 𝐵)) ∈ ℝ)
30	24, 29	eqeltrd 2832	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴𝐷𝐵) ∈ ℝ)
31	23, 30	impbid1 224	1 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ≠ 𝐵) → ((𝐴𝐷𝐵) ∈ ℝ ↔ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ifcif 4506 class class class wbr 5125 ‘cfv 6516 (class class class)co 7377 ℂcc 11073 ℝcr 11074 ℝ^*cxr 11212 ≤ cle 11214 − cmin 11409 -_𝑒cxne 13054 +_𝑒 cxad 13055 abscabs 15146 distcds 17171 ℝ^*_𝑠cxrs 17411

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8670 df-en 8906 df-dom 8907 df-sdom 8908 df-fin 8909 df-sup 9402 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-div 11837 df-nn 12178 df-2 12240 df-3 12241 df-4 12242 df-5 12243 df-6 12244 df-7 12245 df-8 12246 df-9 12247 df-n0 12438 df-z 12524 df-dec 12643 df-uz 12788 df-rp 12940 df-xneg 13057 df-xadd 13058 df-fz 13450 df-seq 13932 df-exp 13993 df-cj 15011 df-re 15012 df-im 15013 df-sqrt 15147 df-abs 15148 df-struct 17045 df-slot 17080 df-ndx 17092 df-base 17110 df-plusg 17175 df-mulr 17176 df-tset 17181 df-ple 17182 df-ds 17184 df-xrs 17413

This theorem is referenced by: xrsxmet 24224 xrsblre 24226 xrsmopn 24227

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