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Mirrors > Home > HSE Home > Th. List > pjnorm2 | Structured version Visualization version GIF version |
Description: A vector belongs to the subspace of a projection iff the norm of its projection equals its norm. This and pjch 30042 yield Theorem 26.3 of [Halmos] p. 44. (Contributed by NM, 7-Apr-2001.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjnorm2 | ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ 𝐻 ↔ (normℎ‘((projℎ‘𝐻)‘𝐴)) = (normℎ‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjhcl 29749 | . . . 4 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((projℎ‘𝐻)‘𝐴) ∈ ℋ) | |
2 | normcl 29473 | . . . 4 ⊢ (((projℎ‘𝐻)‘𝐴) ∈ ℋ → (normℎ‘((projℎ‘𝐻)‘𝐴)) ∈ ℝ) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (normℎ‘((projℎ‘𝐻)‘𝐴)) ∈ ℝ) |
4 | normcl 29473 | . . . 4 ⊢ (𝐴 ∈ ℋ → (normℎ‘𝐴) ∈ ℝ) | |
5 | 4 | adantl 482 | . . 3 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (normℎ‘𝐴) ∈ ℝ) |
6 | 3, 5 | eqleltd 11107 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ((normℎ‘((projℎ‘𝐻)‘𝐴)) = (normℎ‘𝐴) ↔ ((normℎ‘((projℎ‘𝐻)‘𝐴)) ≤ (normℎ‘𝐴) ∧ ¬ (normℎ‘((projℎ‘𝐻)‘𝐴)) < (normℎ‘𝐴)))) |
7 | pjnorm 30072 | . . 3 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (normℎ‘((projℎ‘𝐻)‘𝐴)) ≤ (normℎ‘𝐴)) | |
8 | 7 | biantrurd 533 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (¬ (normℎ‘((projℎ‘𝐻)‘𝐴)) < (normℎ‘𝐴) ↔ ((normℎ‘((projℎ‘𝐻)‘𝐴)) ≤ (normℎ‘𝐴) ∧ ¬ (normℎ‘((projℎ‘𝐻)‘𝐴)) < (normℎ‘𝐴)))) |
9 | pjnel 30074 | . . 3 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (¬ 𝐴 ∈ 𝐻 ↔ (normℎ‘((projℎ‘𝐻)‘𝐴)) < (normℎ‘𝐴))) | |
10 | 9 | con1bid 356 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (¬ (normℎ‘((projℎ‘𝐻)‘𝐴)) < (normℎ‘𝐴) ↔ 𝐴 ∈ 𝐻)) |
11 | 6, 8, 10 | 3bitr2rd 308 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (𝐴 ∈ 𝐻 ↔ (normℎ‘((projℎ‘𝐻)‘𝐴)) = (normℎ‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 class class class wbr 5074 ‘cfv 6427 ℝcr 10858 < clt 10997 ≤ cle 10998 ℋchba 29267 normℎcno 29271 Cℋ cch 29277 projℎcpjh 29285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 ax-inf2 9387 ax-cc 10179 ax-cnex 10915 ax-resscn 10916 ax-1cn 10917 ax-icn 10918 ax-addcl 10919 ax-addrcl 10920 ax-mulcl 10921 ax-mulrcl 10922 ax-mulcom 10923 ax-addass 10924 ax-mulass 10925 ax-distr 10926 ax-i2m1 10927 ax-1ne0 10928 ax-1rid 10929 ax-rnegex 10930 ax-rrecex 10931 ax-cnre 10932 ax-pre-lttri 10933 ax-pre-lttrn 10934 ax-pre-ltadd 10935 ax-pre-mulgt0 10936 ax-pre-sup 10937 ax-addf 10938 ax-mulf 10939 ax-hilex 29347 ax-hfvadd 29348 ax-hvcom 29349 ax-hvass 29350 ax-hv0cl 29351 ax-hvaddid 29352 ax-hfvmul 29353 ax-hvmulid 29354 ax-hvmulass 29355 ax-hvdistr1 29356 ax-hvdistr2 29357 ax-hvmul0 29358 ax-hfi 29427 ax-his1 29430 ax-his2 29431 ax-his3 29432 ax-his4 29433 ax-hcompl 29550 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5485 df-eprel 5491 df-po 5499 df-so 5500 df-fr 5540 df-se 5541 df-we 5542 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-pred 6196 df-ord 6263 df-on 6264 df-lim 6265 df-suc 6266 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-isom 6436 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7704 df-1st 7821 df-2nd 7822 df-supp 7966 df-frecs 8085 df-wrecs 8116 df-recs 8190 df-rdg 8229 df-1o 8285 df-2o 8286 df-oadd 8289 df-omul 8290 df-er 8486 df-map 8605 df-pm 8606 df-ixp 8674 df-en 8722 df-dom 8723 df-sdom 8724 df-fin 8725 df-fsupp 9117 df-fi 9158 df-sup 9189 df-inf 9190 df-oi 9257 df-card 9685 df-acn 9688 df-pnf 10999 df-mnf 11000 df-xr 11001 df-ltxr 11002 df-le 11003 df-sub 11195 df-neg 11196 df-div 11621 df-nn 11962 df-2 12024 df-3 12025 df-4 12026 df-5 12027 df-6 12028 df-7 12029 df-8 12030 df-9 12031 df-n0 12222 df-z 12308 df-dec 12426 df-uz 12571 df-q 12677 df-rp 12719 df-xneg 12836 df-xadd 12837 df-xmul 12838 df-ioo 13071 df-ico 13073 df-icc 13074 df-fz 13228 df-fzo 13371 df-fl 13500 df-seq 13710 df-exp 13771 df-hash 14033 df-cj 14798 df-re 14799 df-im 14800 df-sqrt 14934 df-abs 14935 df-clim 15185 df-rlim 15186 df-sum 15386 df-struct 16836 df-sets 16853 df-slot 16871 df-ndx 16883 df-base 16901 df-ress 16930 df-plusg 16963 df-mulr 16964 df-starv 16965 df-sca 16966 df-vsca 16967 df-ip 16968 df-tset 16969 df-ple 16970 df-ds 16972 df-unif 16973 df-hom 16974 df-cco 16975 df-rest 17121 df-topn 17122 df-0g 17140 df-gsum 17141 df-topgen 17142 df-pt 17143 df-prds 17146 df-xrs 17201 df-qtop 17206 df-imas 17207 df-xps 17209 df-mre 17283 df-mrc 17284 df-acs 17286 df-mgm 18314 df-sgrp 18363 df-mnd 18374 df-submnd 18419 df-mulg 18689 df-cntz 18911 df-cmn 19376 df-psmet 20577 df-xmet 20578 df-met 20579 df-bl 20580 df-mopn 20581 df-fbas 20582 df-fg 20583 df-cnfld 20586 df-top 22031 df-topon 22048 df-topsp 22070 df-bases 22084 df-cld 22158 df-ntr 22159 df-cls 22160 df-nei 22237 df-cn 22366 df-cnp 22367 df-lm 22368 df-haus 22454 df-tx 22701 df-hmeo 22894 df-fil 22985 df-fm 23077 df-flim 23078 df-flf 23079 df-xms 23461 df-ms 23462 df-tms 23463 df-cfil 24407 df-cau 24408 df-cmet 24409 df-grpo 28841 df-gid 28842 df-ginv 28843 df-gdiv 28844 df-ablo 28893 df-vc 28907 df-nv 28940 df-va 28943 df-ba 28944 df-sm 28945 df-0v 28946 df-vs 28947 df-nmcv 28948 df-ims 28949 df-dip 29049 df-ssp 29070 df-ph 29161 df-cbn 29211 df-hnorm 29316 df-hba 29317 df-hvsub 29319 df-hlim 29320 df-hcau 29321 df-sh 29555 df-ch 29569 df-oc 29600 df-ch0 29601 df-shs 29656 df-pjh 29743 |
This theorem is referenced by: jplem1 30616 |
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