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Mirrors > Home > MPE Home > Th. List > tmsxpsval2 | Structured version Visualization version GIF version |
Description: Value of the product of two metrics. (Contributed by Mario Carneiro, 2-Sep-2015.) |
Ref | Expression |
---|---|
tmsxps.p | ⊢ 𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) |
tmsxps.1 | ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) |
tmsxps.2 | ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌)) |
tmsxpsval.a | ⊢ (𝜑 → 𝐴 ∈ 𝑋) |
tmsxpsval.b | ⊢ (𝜑 → 𝐵 ∈ 𝑌) |
tmsxpsval.c | ⊢ (𝜑 → 𝐶 ∈ 𝑋) |
tmsxpsval.d | ⊢ (𝜑 → 𝐷 ∈ 𝑌) |
Ref | Expression |
---|---|
tmsxpsval2 | ⊢ (𝜑 → (〈𝐴, 𝐵〉𝑃〈𝐶, 𝐷〉) = if((𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐵𝑁𝐷), (𝐴𝑀𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tmsxps.p | . . 3 ⊢ 𝑃 = (dist‘((toMetSp‘𝑀) ×s (toMetSp‘𝑁))) | |
2 | tmsxps.1 | . . 3 ⊢ (𝜑 → 𝑀 ∈ (∞Met‘𝑋)) | |
3 | tmsxps.2 | . . 3 ⊢ (𝜑 → 𝑁 ∈ (∞Met‘𝑌)) | |
4 | tmsxpsval.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑋) | |
5 | tmsxpsval.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑌) | |
6 | tmsxpsval.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ 𝑋) | |
7 | tmsxpsval.d | . . 3 ⊢ (𝜑 → 𝐷 ∈ 𝑌) | |
8 | 1, 2, 3, 4, 5, 6, 7 | tmsxpsval 23151 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉𝑃〈𝐶, 𝐷〉) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < )) |
9 | xrltso 12537 | . . 3 ⊢ < Or ℝ* | |
10 | xmetcl 22944 | . . . 4 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) ∈ ℝ*) | |
11 | 2, 4, 6, 10 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐴𝑀𝐶) ∈ ℝ*) |
12 | xmetcl 22944 | . . . 4 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → (𝐵𝑁𝐷) ∈ ℝ*) | |
13 | 3, 5, 7, 12 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝐵𝑁𝐷) ∈ ℝ*) |
14 | suppr 8938 | . . 3 ⊢ (( < Or ℝ* ∧ (𝐴𝑀𝐶) ∈ ℝ* ∧ (𝐵𝑁𝐷) ∈ ℝ*) → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) = if((𝐵𝑁𝐷) < (𝐴𝑀𝐶), (𝐴𝑀𝐶), (𝐵𝑁𝐷))) | |
15 | 9, 11, 13, 14 | mp3an2i 1462 | . 2 ⊢ (𝜑 → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) = if((𝐵𝑁𝐷) < (𝐴𝑀𝐶), (𝐴𝑀𝐶), (𝐵𝑁𝐷))) |
16 | xrltnle 10711 | . . . . 5 ⊢ (((𝐵𝑁𝐷) ∈ ℝ* ∧ (𝐴𝑀𝐶) ∈ ℝ*) → ((𝐵𝑁𝐷) < (𝐴𝑀𝐶) ↔ ¬ (𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷))) | |
17 | 13, 11, 16 | syl2anc 586 | . . . 4 ⊢ (𝜑 → ((𝐵𝑁𝐷) < (𝐴𝑀𝐶) ↔ ¬ (𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷))) |
18 | 17 | ifbid 4492 | . . 3 ⊢ (𝜑 → if((𝐵𝑁𝐷) < (𝐴𝑀𝐶), (𝐴𝑀𝐶), (𝐵𝑁𝐷)) = if(¬ (𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐴𝑀𝐶), (𝐵𝑁𝐷))) |
19 | ifnot 4520 | . . 3 ⊢ if(¬ (𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐴𝑀𝐶), (𝐵𝑁𝐷)) = if((𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐵𝑁𝐷), (𝐴𝑀𝐶)) | |
20 | 18, 19 | syl6eq 2875 | . 2 ⊢ (𝜑 → if((𝐵𝑁𝐷) < (𝐴𝑀𝐶), (𝐴𝑀𝐶), (𝐵𝑁𝐷)) = if((𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐵𝑁𝐷), (𝐴𝑀𝐶))) |
21 | 8, 15, 20 | 3eqtrd 2863 | 1 ⊢ (𝜑 → (〈𝐴, 𝐵〉𝑃〈𝐶, 𝐷〉) = if((𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐵𝑁𝐷), (𝐴𝑀𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ifcif 4470 {cpr 4572 〈cop 4576 class class class wbr 5069 Or wor 5476 ‘cfv 6358 (class class class)co 7159 supcsup 8907 ℝ*cxr 10677 < clt 10678 ≤ cle 10679 distcds 16577 ×s cxps 16782 ∞Metcxmet 20533 toMetSpctms 22932 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-2o 8106 df-oadd 8109 df-er 8292 df-map 8411 df-ixp 8465 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-fsupp 8837 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-icc 12748 df-fz 12896 df-fzo 13037 df-seq 13373 df-hash 13694 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-hom 16592 df-cco 16593 df-rest 16699 df-topn 16700 df-0g 16718 df-gsum 16719 df-topgen 16720 df-prds 16724 df-xrs 16778 df-imas 16784 df-xps 16786 df-mre 16860 df-mrc 16861 df-acs 16863 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-mulg 18228 df-cntz 18450 df-cmn 18911 df-psmet 20540 df-xmet 20541 df-bl 20543 df-mopn 20544 df-top 21505 df-topon 21522 df-topsp 21544 df-bases 21557 df-xms 22933 df-tms 22935 |
This theorem is referenced by: txmetcnp 23160 |
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