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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 24538 | . 2 ⊢ (𝜑 → (〈𝐴, 𝐵〉𝑃〈𝐶, 𝐷〉) = sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < )) |
9 | xrltso 13174 | . . 3 ⊢ < Or ℝ* | |
10 | xmetcl 24328 | . . . 4 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝐴 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋) → (𝐴𝑀𝐶) ∈ ℝ*) | |
11 | 2, 4, 6, 10 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐴𝑀𝐶) ∈ ℝ*) |
12 | xmetcl 24328 | . . . 4 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝐵 ∈ 𝑌 ∧ 𝐷 ∈ 𝑌) → (𝐵𝑁𝐷) ∈ ℝ*) | |
13 | 3, 5, 7, 12 | syl3anc 1368 | . . 3 ⊢ (𝜑 → (𝐵𝑁𝐷) ∈ ℝ*) |
14 | suppr 9514 | . . 3 ⊢ (( < Or ℝ* ∧ (𝐴𝑀𝐶) ∈ ℝ* ∧ (𝐵𝑁𝐷) ∈ ℝ*) → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) = if((𝐵𝑁𝐷) < (𝐴𝑀𝐶), (𝐴𝑀𝐶), (𝐵𝑁𝐷))) | |
15 | 9, 11, 13, 14 | mp3an2i 1463 | . 2 ⊢ (𝜑 → sup({(𝐴𝑀𝐶), (𝐵𝑁𝐷)}, ℝ*, < ) = if((𝐵𝑁𝐷) < (𝐴𝑀𝐶), (𝐴𝑀𝐶), (𝐵𝑁𝐷))) |
16 | xrltnle 11331 | . . . . 5 ⊢ (((𝐵𝑁𝐷) ∈ ℝ* ∧ (𝐴𝑀𝐶) ∈ ℝ*) → ((𝐵𝑁𝐷) < (𝐴𝑀𝐶) ↔ ¬ (𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷))) | |
17 | 13, 11, 16 | syl2anc 582 | . . . 4 ⊢ (𝜑 → ((𝐵𝑁𝐷) < (𝐴𝑀𝐶) ↔ ¬ (𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷))) |
18 | 17 | ifbid 4556 | . . 3 ⊢ (𝜑 → if((𝐵𝑁𝐷) < (𝐴𝑀𝐶), (𝐴𝑀𝐶), (𝐵𝑁𝐷)) = if(¬ (𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐴𝑀𝐶), (𝐵𝑁𝐷))) |
19 | ifnot 4585 | . . 3 ⊢ if(¬ (𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐴𝑀𝐶), (𝐵𝑁𝐷)) = if((𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐵𝑁𝐷), (𝐴𝑀𝐶)) | |
20 | 18, 19 | eqtrdi 2782 | . 2 ⊢ (𝜑 → if((𝐵𝑁𝐷) < (𝐴𝑀𝐶), (𝐴𝑀𝐶), (𝐵𝑁𝐷)) = if((𝐴𝑀𝐶) ≤ (𝐵𝑁𝐷), (𝐵𝑁𝐷), (𝐴𝑀𝐶))) |
21 | 8, 15, 20 | 3eqtrd 2770 | 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 205 = wceq 1534 ∈ wcel 2099 ifcif 4533 {cpr 4635 〈cop 4639 class class class wbr 5153 Or wor 5593 ‘cfv 6554 (class class class)co 7424 supcsup 9483 ℝ*cxr 11297 < clt 11298 ≤ cle 11299 distcds 17275 ×s cxps 17521 ∞Metcxmet 21328 toMetSpctms 24316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-pre-sup 11236 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-isom 6563 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-of 7690 df-om 7877 df-1st 8003 df-2nd 8004 df-supp 8175 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-2o 8497 df-er 8734 df-map 8857 df-ixp 8927 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-fsupp 9406 df-sup 9485 df-inf 9486 df-oi 9553 df-card 9982 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12611 df-dec 12730 df-uz 12875 df-q 12985 df-rp 13029 df-xneg 13146 df-xadd 13147 df-xmul 13148 df-icc 13385 df-fz 13539 df-fzo 13682 df-seq 14022 df-hash 14348 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-ip 17284 df-tset 17285 df-ple 17286 df-ds 17288 df-hom 17290 df-cco 17291 df-rest 17437 df-topn 17438 df-0g 17456 df-gsum 17457 df-topgen 17458 df-prds 17462 df-xrs 17517 df-imas 17523 df-xps 17525 df-mre 17599 df-mrc 17600 df-acs 17602 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-mulg 19062 df-cntz 19311 df-cmn 19780 df-psmet 21335 df-xmet 21336 df-bl 21338 df-mopn 21339 df-top 22887 df-topon 22904 df-topsp 22926 df-bases 22940 df-xms 24317 df-tms 24319 |
This theorem is referenced by: txmetcnp 24547 |
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