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Mirrors > Home > HSE Home > Th. List > pjssdif1i | Structured version Visualization version GIF version |
Description: A necessary and sufficient condition for the difference between two projectors to be a projector. Part 1 of Theorem 29.3 of [Halmos] p. 48 (shortened with pjssposi 29876). (Contributed by NM, 2-Jun-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjco.1 | ⊢ 𝐺 ∈ Cℋ |
pjco.2 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjssdif1i | ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjco.1 | . . 3 ⊢ 𝐺 ∈ Cℋ | |
2 | pjco.2 | . . 3 ⊢ 𝐻 ∈ Cℋ | |
3 | 1, 2 | pjssdif2i 29878 | . 2 ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) −op (projℎ‘𝐺)) = (projℎ‘(𝐻 ∩ (⊥‘𝐺)))) |
4 | pjmfn 29419 | . . . . 5 ⊢ projℎ Fn Cℋ | |
5 | 1 | choccli 29011 | . . . . . 6 ⊢ (⊥‘𝐺) ∈ Cℋ |
6 | 2, 5 | chincli 29164 | . . . . 5 ⊢ (𝐻 ∩ (⊥‘𝐺)) ∈ Cℋ |
7 | fnfvelrn 6840 | . . . . 5 ⊢ ((projℎ Fn Cℋ ∧ (𝐻 ∩ (⊥‘𝐺)) ∈ Cℋ ) → (projℎ‘(𝐻 ∩ (⊥‘𝐺))) ∈ ran projℎ) | |
8 | 4, 6, 7 | mp2an 688 | . . . 4 ⊢ (projℎ‘(𝐻 ∩ (⊥‘𝐺))) ∈ ran projℎ |
9 | eleq1 2897 | . . . 4 ⊢ (((projℎ‘𝐻) −op (projℎ‘𝐺)) = (projℎ‘(𝐻 ∩ (⊥‘𝐺))) → (((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ ↔ (projℎ‘(𝐻 ∩ (⊥‘𝐺))) ∈ ran projℎ)) | |
10 | 8, 9 | mpbiri 259 | . . 3 ⊢ (((projℎ‘𝐻) −op (projℎ‘𝐺)) = (projℎ‘(𝐻 ∩ (⊥‘𝐺))) → ((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ) |
11 | fvelrnb 6719 | . . . . . 6 ⊢ (projℎ Fn Cℋ → (((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ ↔ ∃𝑥 ∈ Cℋ (projℎ‘𝑥) = ((projℎ‘𝐻) −op (projℎ‘𝐺)))) | |
12 | 4, 11 | ax-mp 5 | . . . . 5 ⊢ (((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ ↔ ∃𝑥 ∈ Cℋ (projℎ‘𝑥) = ((projℎ‘𝐻) −op (projℎ‘𝐺))) |
13 | pjige0 29395 | . . . . . . . . 9 ⊢ ((𝑥 ∈ Cℋ ∧ 𝑦 ∈ ℋ) → 0 ≤ (((projℎ‘𝑥)‘𝑦) ·ih 𝑦)) | |
14 | 13 | adantlr 711 | . . . . . . . 8 ⊢ (((𝑥 ∈ Cℋ ∧ (projℎ‘𝑥) = ((projℎ‘𝐻) −op (projℎ‘𝐺))) ∧ 𝑦 ∈ ℋ) → 0 ≤ (((projℎ‘𝑥)‘𝑦) ·ih 𝑦)) |
15 | fveq1 6662 | . . . . . . . . . . 11 ⊢ ((projℎ‘𝑥) = ((projℎ‘𝐻) −op (projℎ‘𝐺)) → ((projℎ‘𝑥)‘𝑦) = (((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑦)) | |
16 | 15 | oveq1d 7160 | . . . . . . . . . 10 ⊢ ((projℎ‘𝑥) = ((projℎ‘𝐻) −op (projℎ‘𝐺)) → (((projℎ‘𝑥)‘𝑦) ·ih 𝑦) = ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑦) ·ih 𝑦)) |
17 | 16 | breq2d 5069 | . . . . . . . . 9 ⊢ ((projℎ‘𝑥) = ((projℎ‘𝐻) −op (projℎ‘𝐺)) → (0 ≤ (((projℎ‘𝑥)‘𝑦) ·ih 𝑦) ↔ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑦) ·ih 𝑦))) |
18 | 17 | ad2antlr 723 | . . . . . . . 8 ⊢ (((𝑥 ∈ Cℋ ∧ (projℎ‘𝑥) = ((projℎ‘𝐻) −op (projℎ‘𝐺))) ∧ 𝑦 ∈ ℋ) → (0 ≤ (((projℎ‘𝑥)‘𝑦) ·ih 𝑦) ↔ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑦) ·ih 𝑦))) |
19 | 14, 18 | mpbid 233 | . . . . . . 7 ⊢ (((𝑥 ∈ Cℋ ∧ (projℎ‘𝑥) = ((projℎ‘𝐻) −op (projℎ‘𝐺))) ∧ 𝑦 ∈ ℋ) → 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑦) ·ih 𝑦)) |
20 | 19 | ralrimiva 3179 | . . . . . 6 ⊢ ((𝑥 ∈ Cℋ ∧ (projℎ‘𝑥) = ((projℎ‘𝐻) −op (projℎ‘𝐺))) → ∀𝑦 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑦) ·ih 𝑦)) |
21 | 20 | rexlimiva 3278 | . . . . 5 ⊢ (∃𝑥 ∈ Cℋ (projℎ‘𝑥) = ((projℎ‘𝐻) −op (projℎ‘𝐺)) → ∀𝑦 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑦) ·ih 𝑦)) |
22 | 12, 21 | sylbi 218 | . . . 4 ⊢ (((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ → ∀𝑦 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑦) ·ih 𝑦)) |
23 | 1, 2 | pjssposi 29876 | . . . . 5 ⊢ (∀𝑦 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑦) ·ih 𝑦) ↔ 𝐺 ⊆ 𝐻) |
24 | 23, 3 | bitri 276 | . . . 4 ⊢ (∀𝑦 ∈ ℋ 0 ≤ ((((projℎ‘𝐻) −op (projℎ‘𝐺))‘𝑦) ·ih 𝑦) ↔ ((projℎ‘𝐻) −op (projℎ‘𝐺)) = (projℎ‘(𝐻 ∩ (⊥‘𝐺)))) |
25 | 22, 24 | sylib 219 | . . 3 ⊢ (((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ → ((projℎ‘𝐻) −op (projℎ‘𝐺)) = (projℎ‘(𝐻 ∩ (⊥‘𝐺)))) |
26 | 10, 25 | impbii 210 | . 2 ⊢ (((projℎ‘𝐻) −op (projℎ‘𝐺)) = (projℎ‘(𝐻 ∩ (⊥‘𝐺))) ↔ ((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ) |
27 | 3, 26 | bitri 276 | 1 ⊢ (𝐺 ⊆ 𝐻 ↔ ((projℎ‘𝐻) −op (projℎ‘𝐺)) ∈ ran projℎ) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 ∩ cin 3932 ⊆ wss 3933 class class class wbr 5057 ran crn 5549 Fn wfn 6343 ‘cfv 6348 (class class class)co 7145 0cc0 10525 ≤ cle 10664 ℋchba 28623 ·ih csp 28626 Cℋ cch 28633 ⊥cort 28634 projℎcpjh 28641 −op chod 28644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cc 9845 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 ax-hilex 28703 ax-hfvadd 28704 ax-hvcom 28705 ax-hvass 28706 ax-hv0cl 28707 ax-hvaddid 28708 ax-hfvmul 28709 ax-hvmulid 28710 ax-hvmulass 28711 ax-hvdistr1 28712 ax-hvdistr2 28713 ax-hvmul0 28714 ax-hfi 28783 ax-his1 28786 ax-his2 28787 ax-his3 28788 ax-his4 28789 ax-hcompl 28906 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-omul 8096 df-er 8278 df-map 8397 df-pm 8398 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-acn 9359 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-fl 13150 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-rlim 14834 df-sum 15031 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-cn 21763 df-cnp 21764 df-lm 21765 df-haus 21851 df-tx 22098 df-hmeo 22291 df-fil 22382 df-fm 22474 df-flim 22475 df-flf 22476 df-xms 22857 df-ms 22858 df-tms 22859 df-cfil 23785 df-cau 23786 df-cmet 23787 df-grpo 28197 df-gid 28198 df-ginv 28199 df-gdiv 28200 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-vs 28303 df-nmcv 28304 df-ims 28305 df-dip 28405 df-ssp 28426 df-ph 28517 df-cbn 28567 df-hnorm 28672 df-hba 28673 df-hvsub 28675 df-hlim 28676 df-hcau 28677 df-sh 28911 df-ch 28925 df-oc 28956 df-ch0 28957 df-shs 29012 df-pjh 29099 df-hodif 29436 |
This theorem is referenced by: (None) |
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