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Mirrors > Home > HSE Home > Th. List > pjclem3 | Structured version Visualization version GIF version |
Description: Lemma for projection commutation theorem. (Contributed by NM, 26-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjclem1.1 | ⊢ 𝐺 ∈ Cℋ |
pjclem1.2 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pjclem3 | ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻))) = ((projℎ‘(⊥‘𝐻)) ∘ (projℎ‘𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iop 30090 | . . . . . . . 8 ⊢ Iop = (projℎ‘ ℋ) | |
2 | 1 | coeq2i 5766 | . . . . . . 7 ⊢ ((projℎ‘𝐺) ∘ Iop ) = ((projℎ‘𝐺) ∘ (projℎ‘ ℋ)) |
3 | pjclem1.1 | . . . . . . . . 9 ⊢ 𝐺 ∈ Cℋ | |
4 | 3 | pjfi 30045 | . . . . . . . 8 ⊢ (projℎ‘𝐺): ℋ⟶ ℋ |
5 | 4 | hoid1i 30130 | . . . . . . 7 ⊢ ((projℎ‘𝐺) ∘ Iop ) = (projℎ‘𝐺) |
6 | 2, 5 | eqtr3i 2769 | . . . . . 6 ⊢ ((projℎ‘𝐺) ∘ (projℎ‘ ℋ)) = (projℎ‘𝐺) |
7 | 4 | hoid1ri 30131 | . . . . . 6 ⊢ ( Iop ∘ (projℎ‘𝐺)) = (projℎ‘𝐺) |
8 | 1 | coeq1i 5765 | . . . . . 6 ⊢ ( Iop ∘ (projℎ‘𝐺)) = ((projℎ‘ ℋ) ∘ (projℎ‘𝐺)) |
9 | 6, 7, 8 | 3eqtr2i 2773 | . . . . 5 ⊢ ((projℎ‘𝐺) ∘ (projℎ‘ ℋ)) = ((projℎ‘ ℋ) ∘ (projℎ‘𝐺)) |
10 | 9 | oveq1i 7278 | . . . 4 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘ ℋ)) −op ((projℎ‘𝐺) ∘ (projℎ‘𝐻))) = (((projℎ‘ ℋ) ∘ (projℎ‘𝐺)) −op ((projℎ‘𝐺) ∘ (projℎ‘𝐻))) |
11 | oveq2 7276 | . . . 4 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → (((projℎ‘ ℋ) ∘ (projℎ‘𝐺)) −op ((projℎ‘𝐺) ∘ (projℎ‘𝐻))) = (((projℎ‘ ℋ) ∘ (projℎ‘𝐺)) −op ((projℎ‘𝐻) ∘ (projℎ‘𝐺)))) | |
12 | 10, 11 | eqtrid 2791 | . . 3 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → (((projℎ‘𝐺) ∘ (projℎ‘ ℋ)) −op ((projℎ‘𝐺) ∘ (projℎ‘𝐻))) = (((projℎ‘ ℋ) ∘ (projℎ‘𝐺)) −op ((projℎ‘𝐻) ∘ (projℎ‘𝐺)))) |
13 | helch 29584 | . . . . 5 ⊢ ℋ ∈ Cℋ | |
14 | 13 | pjfi 30045 | . . . 4 ⊢ (projℎ‘ ℋ): ℋ⟶ ℋ |
15 | pjclem1.2 | . . . . 5 ⊢ 𝐻 ∈ Cℋ | |
16 | 15 | pjfi 30045 | . . . 4 ⊢ (projℎ‘𝐻): ℋ⟶ ℋ |
17 | 3, 14, 16 | pjddii 30497 | . . 3 ⊢ ((projℎ‘𝐺) ∘ ((projℎ‘ ℋ) −op (projℎ‘𝐻))) = (((projℎ‘𝐺) ∘ (projℎ‘ ℋ)) −op ((projℎ‘𝐺) ∘ (projℎ‘𝐻))) |
18 | 14, 16, 4 | hocsubdiri 30121 | . . 3 ⊢ (((projℎ‘ ℋ) −op (projℎ‘𝐻)) ∘ (projℎ‘𝐺)) = (((projℎ‘ ℋ) ∘ (projℎ‘𝐺)) −op ((projℎ‘𝐻) ∘ (projℎ‘𝐺))) |
19 | 12, 17, 18 | 3eqtr4g 2804 | . 2 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ ((projℎ‘ ℋ) −op (projℎ‘𝐻))) = (((projℎ‘ ℋ) −op (projℎ‘𝐻)) ∘ (projℎ‘𝐺))) |
20 | 15 | pjoci 30521 | . . 3 ⊢ ((projℎ‘ ℋ) −op (projℎ‘𝐻)) = (projℎ‘(⊥‘𝐻)) |
21 | 20 | coeq2i 5766 | . 2 ⊢ ((projℎ‘𝐺) ∘ ((projℎ‘ ℋ) −op (projℎ‘𝐻))) = ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻))) |
22 | 20 | coeq1i 5765 | . 2 ⊢ (((projℎ‘ ℋ) −op (projℎ‘𝐻)) ∘ (projℎ‘𝐺)) = ((projℎ‘(⊥‘𝐻)) ∘ (projℎ‘𝐺)) |
23 | 19, 21, 22 | 3eqtr3g 2802 | 1 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐻) ∘ (projℎ‘𝐺)) → ((projℎ‘𝐺) ∘ (projℎ‘(⊥‘𝐻))) = ((projℎ‘(⊥‘𝐻)) ∘ (projℎ‘𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2109 ∘ ccom 5592 ‘cfv 6430 (class class class)co 7268 ℋchba 29260 Cℋ cch 29270 ⊥cort 29271 projℎcpjh 29278 −op chod 29281 Iop chio 29285 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-inf2 9360 ax-cc 10175 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-pre-sup 10933 ax-addf 10934 ax-mulf 10935 ax-hilex 29340 ax-hfvadd 29341 ax-hvcom 29342 ax-hvass 29343 ax-hv0cl 29344 ax-hvaddid 29345 ax-hfvmul 29346 ax-hvmulid 29347 ax-hvmulass 29348 ax-hvdistr1 29349 ax-hvdistr2 29350 ax-hvmul0 29351 ax-hfi 29420 ax-his1 29423 ax-his2 29424 ax-his3 29425 ax-his4 29426 ax-hcompl 29543 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-se 5544 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-isom 6439 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-of 7524 df-om 7701 df-1st 7817 df-2nd 7818 df-supp 7962 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-2o 8282 df-oadd 8285 df-omul 8286 df-er 8472 df-map 8591 df-pm 8592 df-ixp 8660 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-fsupp 9090 df-fi 9131 df-sup 9162 df-inf 9163 df-oi 9230 df-card 9681 df-acn 9684 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-div 11616 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-7 12024 df-8 12025 df-9 12026 df-n0 12217 df-z 12303 df-dec 12420 df-uz 12565 df-q 12671 df-rp 12713 df-xneg 12830 df-xadd 12831 df-xmul 12832 df-ioo 13065 df-ico 13067 df-icc 13068 df-fz 13222 df-fzo 13365 df-fl 13493 df-seq 13703 df-exp 13764 df-hash 14026 df-cj 14791 df-re 14792 df-im 14793 df-sqrt 14927 df-abs 14928 df-clim 15178 df-rlim 15179 df-sum 15379 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-starv 16958 df-sca 16959 df-vsca 16960 df-ip 16961 df-tset 16962 df-ple 16963 df-ds 16965 df-unif 16966 df-hom 16967 df-cco 16968 df-rest 17114 df-topn 17115 df-0g 17133 df-gsum 17134 df-topgen 17135 df-pt 17136 df-prds 17139 df-xrs 17194 df-qtop 17199 df-imas 17200 df-xps 17202 df-mre 17276 df-mrc 17277 df-acs 17279 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-submnd 18412 df-mulg 18682 df-cntz 18904 df-cmn 19369 df-psmet 20570 df-xmet 20571 df-met 20572 df-bl 20573 df-mopn 20574 df-fbas 20575 df-fg 20576 df-cnfld 20579 df-top 22024 df-topon 22041 df-topsp 22063 df-bases 22077 df-cld 22151 df-ntr 22152 df-cls 22153 df-nei 22230 df-cn 22359 df-cnp 22360 df-lm 22361 df-haus 22447 df-tx 22694 df-hmeo 22887 df-fil 22978 df-fm 23070 df-flim 23071 df-flf 23072 df-xms 23454 df-ms 23455 df-tms 23456 df-cfil 24400 df-cau 24401 df-cmet 24402 df-grpo 28834 df-gid 28835 df-ginv 28836 df-gdiv 28837 df-ablo 28886 df-vc 28900 df-nv 28933 df-va 28936 df-ba 28937 df-sm 28938 df-0v 28939 df-vs 28940 df-nmcv 28941 df-ims 28942 df-dip 29042 df-ssp 29063 df-ph 29154 df-cbn 29204 df-hnorm 29309 df-hba 29310 df-hvsub 29312 df-hlim 29313 df-hcau 29314 df-sh 29548 df-ch 29562 df-oc 29593 df-ch0 29594 df-shs 29649 df-pjh 29736 df-hosum 30071 df-hodif 30073 df-iop 30090 |
This theorem is referenced by: pjci 30541 |
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