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Mirrors > Home > HSE Home > Th. List > pjsdi2i | Structured version Visualization version GIF version |
Description: Chained distributive law for Hilbert space operator difference. (Contributed by NM, 30-Nov-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjsdi2.1 | ⊢ 𝐻 ∈ Cℋ |
pjsdi2.2 | ⊢ 𝑅: ℋ⟶ ℋ |
pjsdi2.3 | ⊢ 𝑆: ℋ⟶ ℋ |
pjsdi2.4 | ⊢ 𝑇: ℋ⟶ ℋ |
Ref | Expression |
---|---|
pjsdi2i | ⊢ ((𝑅 ∘ (𝑆 +op 𝑇)) = ((𝑅 ∘ 𝑆) +op (𝑅 ∘ 𝑇)) → (((projℎ‘𝐻) ∘ 𝑅) ∘ (𝑆 +op 𝑇)) = ((((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑆) +op (((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑇))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq2 5861 | . . 3 ⊢ ((𝑅 ∘ (𝑆 +op 𝑇)) = ((𝑅 ∘ 𝑆) +op (𝑅 ∘ 𝑇)) → ((projℎ‘𝐻) ∘ (𝑅 ∘ (𝑆 +op 𝑇))) = ((projℎ‘𝐻) ∘ ((𝑅 ∘ 𝑆) +op (𝑅 ∘ 𝑇)))) | |
2 | pjsdi2.1 | . . . 4 ⊢ 𝐻 ∈ Cℋ | |
3 | pjsdi2.2 | . . . . 5 ⊢ 𝑅: ℋ⟶ ℋ | |
4 | pjsdi2.3 | . . . . 5 ⊢ 𝑆: ℋ⟶ ℋ | |
5 | 3, 4 | hocofi 31668 | . . . 4 ⊢ (𝑅 ∘ 𝑆): ℋ⟶ ℋ |
6 | pjsdi2.4 | . . . . 5 ⊢ 𝑇: ℋ⟶ ℋ | |
7 | 3, 6 | hocofi 31668 | . . . 4 ⊢ (𝑅 ∘ 𝑇): ℋ⟶ ℋ |
8 | 2, 5, 7 | pjsdii 32057 | . . 3 ⊢ ((projℎ‘𝐻) ∘ ((𝑅 ∘ 𝑆) +op (𝑅 ∘ 𝑇))) = (((projℎ‘𝐻) ∘ (𝑅 ∘ 𝑆)) +op ((projℎ‘𝐻) ∘ (𝑅 ∘ 𝑇))) |
9 | 1, 8 | eqtrdi 2781 | . 2 ⊢ ((𝑅 ∘ (𝑆 +op 𝑇)) = ((𝑅 ∘ 𝑆) +op (𝑅 ∘ 𝑇)) → ((projℎ‘𝐻) ∘ (𝑅 ∘ (𝑆 +op 𝑇))) = (((projℎ‘𝐻) ∘ (𝑅 ∘ 𝑆)) +op ((projℎ‘𝐻) ∘ (𝑅 ∘ 𝑇)))) |
10 | coass 6271 | . 2 ⊢ (((projℎ‘𝐻) ∘ 𝑅) ∘ (𝑆 +op 𝑇)) = ((projℎ‘𝐻) ∘ (𝑅 ∘ (𝑆 +op 𝑇))) | |
11 | coass 6271 | . . 3 ⊢ (((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑆) = ((projℎ‘𝐻) ∘ (𝑅 ∘ 𝑆)) | |
12 | coass 6271 | . . 3 ⊢ (((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑇) = ((projℎ‘𝐻) ∘ (𝑅 ∘ 𝑇)) | |
13 | 11, 12 | oveq12i 7431 | . 2 ⊢ ((((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑆) +op (((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑇)) = (((projℎ‘𝐻) ∘ (𝑅 ∘ 𝑆)) +op ((projℎ‘𝐻) ∘ (𝑅 ∘ 𝑇))) |
14 | 9, 10, 13 | 3eqtr4g 2790 | 1 ⊢ ((𝑅 ∘ (𝑆 +op 𝑇)) = ((𝑅 ∘ 𝑆) +op (𝑅 ∘ 𝑇)) → (((projℎ‘𝐻) ∘ 𝑅) ∘ (𝑆 +op 𝑇)) = ((((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑆) +op (((projℎ‘𝐻) ∘ 𝑅) ∘ 𝑇))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ∘ ccom 5682 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℋchba 30821 Cℋ cch 30831 projℎcpjh 30839 +op chos 30840 |
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 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-inf2 9671 ax-cc 10465 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 ax-addf 11224 ax-mulf 11225 ax-hilex 30901 ax-hfvadd 30902 ax-hvcom 30903 ax-hvass 30904 ax-hv0cl 30905 ax-hvaddid 30906 ax-hfvmul 30907 ax-hvmulid 30908 ax-hvmulass 30909 ax-hvdistr1 30910 ax-hvdistr2 30911 ax-hvmul0 30912 ax-hfi 30981 ax-his1 30984 ax-his2 30985 ax-his3 30986 ax-his4 30987 ax-hcompl 31104 |
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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-oadd 8491 df-omul 8492 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9393 df-fi 9441 df-sup 9472 df-inf 9473 df-oi 9540 df-card 9969 df-acn 9972 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ioo 13368 df-ico 13370 df-icc 13371 df-fz 13525 df-fzo 13668 df-fl 13798 df-seq 14008 df-exp 14068 df-hash 14334 df-cj 15090 df-re 15091 df-im 15092 df-sqrt 15226 df-abs 15227 df-clim 15476 df-rlim 15477 df-sum 15677 df-struct 17135 df-sets 17152 df-slot 17170 df-ndx 17182 df-base 17200 df-ress 17229 df-plusg 17265 df-mulr 17266 df-starv 17267 df-sca 17268 df-vsca 17269 df-ip 17270 df-tset 17271 df-ple 17272 df-ds 17274 df-unif 17275 df-hom 17276 df-cco 17277 df-rest 17423 df-topn 17424 df-0g 17442 df-gsum 17443 df-topgen 17444 df-pt 17445 df-prds 17448 df-xrs 17503 df-qtop 17508 df-imas 17509 df-xps 17511 df-mre 17585 df-mrc 17586 df-acs 17588 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-submnd 18760 df-mulg 19048 df-cntz 19297 df-cmn 19766 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22857 df-topon 22874 df-topsp 22896 df-bases 22910 df-cld 22984 df-ntr 22985 df-cls 22986 df-nei 23063 df-cn 23192 df-cnp 23193 df-lm 23194 df-haus 23280 df-tx 23527 df-hmeo 23720 df-fil 23811 df-fm 23903 df-flim 23904 df-flf 23905 df-xms 24287 df-ms 24288 df-tms 24289 df-cfil 25244 df-cau 25245 df-cmet 25246 df-grpo 30395 df-gid 30396 df-ginv 30397 df-gdiv 30398 df-ablo 30447 df-vc 30461 df-nv 30494 df-va 30497 df-ba 30498 df-sm 30499 df-0v 30500 df-vs 30501 df-nmcv 30502 df-ims 30503 df-dip 30603 df-ssp 30624 df-ph 30715 df-cbn 30765 df-hnorm 30870 df-hba 30871 df-hvsub 30873 df-hlim 30874 df-hcau 30875 df-sh 31109 df-ch 31123 df-oc 31154 df-ch0 31155 df-shs 31210 df-pjh 31297 df-hosum 31632 |
This theorem is referenced by: (None) |
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