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Mirrors > Home > HSE Home > Th. List > pjssge0ii | Structured version Visualization version GIF version |
Description: Theorem 4.5(iv)->(v) of [Beran] p. 112. (Contributed by NM, 13-Aug-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjidm.1 | โข ๐ป โ Cโ |
pjidm.2 | โข ๐ด โ โ |
pjsslem.1 | โข ๐บ โ Cโ |
Ref | Expression |
---|---|
pjssge0ii | โข ((((projโโ๐บ)โ๐ด) โโ ((projโโ๐ป)โ๐ด)) = ((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด) โ 0 โค ((((projโโ๐บ)โ๐ด) โโ ((projโโ๐ป)โ๐ด)) ยทih ๐ด)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjsslem.1 | . . . . . 6 โข ๐บ โ Cโ | |
2 | pjidm.1 | . . . . . . 7 โข ๐ป โ Cโ | |
3 | 2 | choccli 31161 | . . . . . 6 โข (โฅโ๐ป) โ Cโ |
4 | 1, 3 | chincli 31314 | . . . . 5 โข (๐บ โฉ (โฅโ๐ป)) โ Cโ |
5 | pjidm.2 | . . . . 5 โข ๐ด โ โ | |
6 | 4, 5 | pjhclii 31276 | . . . 4 โข ((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด) โ โ |
7 | 6 | normcli 30985 | . . 3 โข (normโโ((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด)) โ โ |
8 | 7 | sqge0i 14183 | . 2 โข 0 โค ((normโโ((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด))โ2) |
9 | oveq1 7423 | . . 3 โข ((((projโโ๐บ)โ๐ด) โโ ((projโโ๐ป)โ๐ด)) = ((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด) โ ((((projโโ๐บ)โ๐ด) โโ ((projโโ๐ป)โ๐ด)) ยทih ๐ด) = (((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด) ยทih ๐ด)) | |
10 | 4, 5 | pjinormii 31530 | . . 3 โข (((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด) ยทih ๐ด) = ((normโโ((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด))โ2) |
11 | 9, 10 | eqtrdi 2781 | . 2 โข ((((projโโ๐บ)โ๐ด) โโ ((projโโ๐ป)โ๐ด)) = ((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด) โ ((((projโโ๐บ)โ๐ด) โโ ((projโโ๐ป)โ๐ด)) ยทih ๐ด) = ((normโโ((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด))โ2)) |
12 | 8, 11 | breqtrrid 5181 | 1 โข ((((projโโ๐บ)โ๐ด) โโ ((projโโ๐ป)โ๐ด)) = ((projโโ(๐บ โฉ (โฅโ๐ป)))โ๐ด) โ 0 โค ((((projโโ๐บ)โ๐ด) โโ ((projโโ๐ป)โ๐ด)) ยทih ๐ด)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โฉ cin 3938 class class class wbr 5143 โcfv 6543 (class class class)co 7416 0cc0 11138 โค cle 11279 2c2 12297 โcexp 14058 โchba 30773 ยทih csp 30776 normโcno 30777 โโ cmv 30779 Cโ cch 30783 โฅcort 30784 projโcpjh 30791 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cc 10458 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 ax-hilex 30853 ax-hfvadd 30854 ax-hvcom 30855 ax-hvass 30856 ax-hv0cl 30857 ax-hvaddid 30858 ax-hfvmul 30859 ax-hvmulid 30860 ax-hvmulass 30861 ax-hvdistr1 30862 ax-hvdistr2 30863 ax-hvmul0 30864 ax-hfi 30933 ax-his1 30936 ax-his2 30937 ax-his3 30938 ax-his4 30939 ax-hcompl 31056 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-seq 13999 df-exp 14059 df-hash 14322 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-clim 15464 df-rlim 15465 df-sum 15665 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18740 df-mulg 19028 df-cntz 19272 df-cmn 19741 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-cn 23149 df-cnp 23150 df-lm 23151 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cfil 25201 df-cau 25202 df-cmet 25203 df-grpo 30347 df-gid 30348 df-ginv 30349 df-gdiv 30350 df-ablo 30399 df-vc 30413 df-nv 30446 df-va 30449 df-ba 30450 df-sm 30451 df-0v 30452 df-vs 30453 df-nmcv 30454 df-ims 30455 df-dip 30555 df-ssp 30576 df-ph 30667 df-cbn 30717 df-hnorm 30822 df-hba 30823 df-hvsub 30825 df-hlim 30826 df-hcau 30827 df-sh 31061 df-ch 31075 df-oc 31106 df-ch0 31107 df-shs 31162 df-pjh 31249 |
This theorem is referenced by: pjssge0i 32020 |
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