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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rmyeq0 | Structured version Visualization version GIF version |
Description: Y is zero only at zero. (Contributed by Stefan O'Rear, 26-Sep-2014.) |
Ref | Expression |
---|---|
rmyeq0 | β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π = 0 β (π΄ Yrm π) = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0z 12570 | . . 3 β’ 0 β β€ | |
2 | oveq2 7412 | . . . 4 β’ (π = π β (π΄ Yrm π) = (π΄ Yrm π)) | |
3 | oveq2 7412 | . . . 4 β’ (π = π β (π΄ Yrm π) = (π΄ Yrm π)) | |
4 | oveq2 7412 | . . . 4 β’ (π = 0 β (π΄ Yrm π) = (π΄ Yrm 0)) | |
5 | zssre 12566 | . . . 4 β’ β€ β β | |
6 | frmy 42212 | . . . . . 6 β’ Yrm :((β€β₯β2) Γ β€)βΆβ€ | |
7 | 6 | fovcl 7532 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm π) β β€) |
8 | 7 | zred 12667 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm π) β β) |
9 | ltrmy 42250 | . . . . . 6 β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π < π β (π΄ Yrm π) < (π΄ Yrm π))) | |
10 | 9 | biimpd 228 | . . . . 5 β’ ((π΄ β (β€β₯β2) β§ π β β€ β§ π β β€) β (π < π β (π΄ Yrm π) < (π΄ Yrm π))) |
11 | 10 | 3expb 1117 | . . . 4 β’ ((π΄ β (β€β₯β2) β§ (π β β€ β§ π β β€)) β (π < π β (π΄ Yrm π) < (π΄ Yrm π))) |
12 | 2, 3, 4, 5, 8, 11 | eqord1 11743 | . . 3 β’ ((π΄ β (β€β₯β2) β§ (π β β€ β§ 0 β β€)) β (π = 0 β (π΄ Yrm π) = (π΄ Yrm 0))) |
13 | 1, 12 | mpanr2 701 | . 2 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π = 0 β (π΄ Yrm π) = (π΄ Yrm 0))) |
14 | rmy0 42227 | . . . 4 β’ (π΄ β (β€β₯β2) β (π΄ Yrm 0) = 0) | |
15 | 14 | adantr 480 | . . 3 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π΄ Yrm 0) = 0) |
16 | 15 | eqeq2d 2737 | . 2 β’ ((π΄ β (β€β₯β2) β§ π β β€) β ((π΄ Yrm π) = (π΄ Yrm 0) β (π΄ Yrm π) = 0)) |
17 | 13, 16 | bitrd 279 | 1 β’ ((π΄ β (β€β₯β2) β§ π β β€) β (π = 0 β (π΄ Yrm π) = 0)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1084 = wceq 1533 β wcel 2098 class class class wbr 5141 βcfv 6536 (class class class)co 7404 0cc0 11109 < clt 11249 2c2 12268 β€cz 12559 β€β₯cuz 12823 Yrm crmy 42198 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8144 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-2o 8465 df-oadd 8468 df-omul 8469 df-er 8702 df-map 8821 df-pm 8822 df-ixp 8891 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-fsupp 9361 df-fi 9405 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-acn 9936 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-xnn0 12546 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ioo 13331 df-ioc 13332 df-ico 13333 df-icc 13334 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14031 df-fac 14237 df-bc 14266 df-hash 14294 df-shft 15018 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-limsup 15419 df-clim 15436 df-rlim 15437 df-sum 15637 df-ef 16015 df-sin 16017 df-cos 16018 df-pi 16020 df-dvds 16203 df-gcd 16441 df-numer 16678 df-denom 16679 df-struct 17087 df-sets 17104 df-slot 17122 df-ndx 17134 df-base 17152 df-ress 17181 df-plusg 17217 df-mulr 17218 df-starv 17219 df-sca 17220 df-vsca 17221 df-ip 17222 df-tset 17223 df-ple 17224 df-ds 17226 df-unif 17227 df-hom 17228 df-cco 17229 df-rest 17375 df-topn 17376 df-0g 17394 df-gsum 17395 df-topgen 17396 df-pt 17397 df-prds 17400 df-xrs 17455 df-qtop 17460 df-imas 17461 df-xps 17463 df-mre 17537 df-mrc 17538 df-acs 17540 df-mgm 18571 df-sgrp 18650 df-mnd 18666 df-submnd 18712 df-mulg 18994 df-cntz 19231 df-cmn 19700 df-psmet 21228 df-xmet 21229 df-met 21230 df-bl 21231 df-mopn 21232 df-fbas 21233 df-fg 21234 df-cnfld 21237 df-top 22747 df-topon 22764 df-topsp 22786 df-bases 22800 df-cld 22874 df-ntr 22875 df-cls 22876 df-nei 22953 df-lp 22991 df-perf 22992 df-cn 23082 df-cnp 23083 df-haus 23170 df-tx 23417 df-hmeo 23610 df-fil 23701 df-fm 23793 df-flim 23794 df-flf 23795 df-xms 24177 df-ms 24178 df-tms 24179 df-cncf 24749 df-limc 25746 df-dv 25747 df-log 26441 df-squarenn 42138 df-pell1qr 42139 df-pell14qr 42140 df-pell1234qr 42141 df-pellfund 42142 df-rmx 42199 df-rmy 42200 |
This theorem is referenced by: jm2.19 42291 |
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