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Mirrors > Home > HSE Home > Th. List > pjaddii | Structured version Visualization version GIF version |
Description: Projection of vector sum is sum of projections. (Contributed by NM, 31-Oct-1999.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjidm.1 | ⊢ 𝐻 ∈ Cℋ |
pjidm.2 | ⊢ 𝐴 ∈ ℋ |
pjadj.3 | ⊢ 𝐵 ∈ ℋ |
Ref | Expression |
---|---|
pjaddii | ⊢ ((projℎ‘𝐻)‘(𝐴 +ℎ 𝐵)) = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pjidm.1 | . . . . . 6 ⊢ 𝐻 ∈ Cℋ | |
2 | pjidm.2 | . . . . . 6 ⊢ 𝐴 ∈ ℋ | |
3 | 1, 2 | pjpji 30604 | . . . . 5 ⊢ 𝐴 = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐴)) |
4 | pjadj.3 | . . . . . 6 ⊢ 𝐵 ∈ ℋ | |
5 | 1, 4 | pjpji 30604 | . . . . 5 ⊢ 𝐵 = (((projℎ‘𝐻)‘𝐵) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵)) |
6 | 3, 5 | oveq12i 7406 | . . . 4 ⊢ (𝐴 +ℎ 𝐵) = ((((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐴)) +ℎ (((projℎ‘𝐻)‘𝐵) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵))) |
7 | 1, 2 | pjhclii 30602 | . . . . 5 ⊢ ((projℎ‘𝐻)‘𝐴) ∈ ℋ |
8 | 1 | choccli 30487 | . . . . . 6 ⊢ (⊥‘𝐻) ∈ Cℋ |
9 | 8, 2 | pjhclii 30602 | . . . . 5 ⊢ ((projℎ‘(⊥‘𝐻))‘𝐴) ∈ ℋ |
10 | 1, 4 | pjhclii 30602 | . . . . 5 ⊢ ((projℎ‘𝐻)‘𝐵) ∈ ℋ |
11 | 8, 4 | pjhclii 30602 | . . . . 5 ⊢ ((projℎ‘(⊥‘𝐻))‘𝐵) ∈ ℋ |
12 | 7, 9, 10, 11 | hvadd4i 30238 | . . . 4 ⊢ ((((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐴)) +ℎ (((projℎ‘𝐻)‘𝐵) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵))) = ((((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) +ℎ (((projℎ‘(⊥‘𝐻))‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵))) |
13 | 6, 12 | eqtri 2760 | . . 3 ⊢ (𝐴 +ℎ 𝐵) = ((((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) +ℎ (((projℎ‘(⊥‘𝐻))‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵))) |
14 | 13 | fveq2i 6882 | . 2 ⊢ ((projℎ‘𝐻)‘(𝐴 +ℎ 𝐵)) = ((projℎ‘𝐻)‘((((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) +ℎ (((projℎ‘(⊥‘𝐻))‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵)))) |
15 | 1 | chshii 30407 | . . . 4 ⊢ 𝐻 ∈ Sℋ |
16 | 1, 2 | pjclii 30601 | . . . 4 ⊢ ((projℎ‘𝐻)‘𝐴) ∈ 𝐻 |
17 | 1, 4 | pjclii 30601 | . . . 4 ⊢ ((projℎ‘𝐻)‘𝐵) ∈ 𝐻 |
18 | shaddcl 30397 | . . . 4 ⊢ ((𝐻 ∈ Sℋ ∧ ((projℎ‘𝐻)‘𝐴) ∈ 𝐻 ∧ ((projℎ‘𝐻)‘𝐵) ∈ 𝐻) → (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) ∈ 𝐻) | |
19 | 15, 16, 17, 18 | mp3an 1461 | . . 3 ⊢ (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) ∈ 𝐻 |
20 | 8 | chshii 30407 | . . . 4 ⊢ (⊥‘𝐻) ∈ Sℋ |
21 | 8, 2 | pjclii 30601 | . . . 4 ⊢ ((projℎ‘(⊥‘𝐻))‘𝐴) ∈ (⊥‘𝐻) |
22 | 8, 4 | pjclii 30601 | . . . 4 ⊢ ((projℎ‘(⊥‘𝐻))‘𝐵) ∈ (⊥‘𝐻) |
23 | shaddcl 30397 | . . . 4 ⊢ (((⊥‘𝐻) ∈ Sℋ ∧ ((projℎ‘(⊥‘𝐻))‘𝐴) ∈ (⊥‘𝐻) ∧ ((projℎ‘(⊥‘𝐻))‘𝐵) ∈ (⊥‘𝐻)) → (((projℎ‘(⊥‘𝐻))‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵)) ∈ (⊥‘𝐻)) | |
24 | 20, 21, 22, 23 | mp3an 1461 | . . 3 ⊢ (((projℎ‘(⊥‘𝐻))‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵)) ∈ (⊥‘𝐻) |
25 | 1 | pjcompi 30852 | . . 3 ⊢ (((((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) ∈ 𝐻 ∧ (((projℎ‘(⊥‘𝐻))‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵)) ∈ (⊥‘𝐻)) → ((projℎ‘𝐻)‘((((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) +ℎ (((projℎ‘(⊥‘𝐻))‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵)))) = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵))) |
26 | 19, 24, 25 | mp2an 690 | . 2 ⊢ ((projℎ‘𝐻)‘((((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) +ℎ (((projℎ‘(⊥‘𝐻))‘𝐴) +ℎ ((projℎ‘(⊥‘𝐻))‘𝐵)))) = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) |
27 | 14, 26 | eqtri 2760 | 1 ⊢ ((projℎ‘𝐻)‘(𝐴 +ℎ 𝐵)) = (((projℎ‘𝐻)‘𝐴) +ℎ ((projℎ‘𝐻)‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ‘cfv 6533 (class class class)co 7394 ℋchba 30099 +ℎ cva 30100 Sℋ csh 30108 Cℋ cch 30109 ⊥cort 30110 projℎcpjh 30117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 ax-inf2 9620 ax-cc 10414 ax-cnex 11150 ax-resscn 11151 ax-1cn 11152 ax-icn 11153 ax-addcl 11154 ax-addrcl 11155 ax-mulcl 11156 ax-mulrcl 11157 ax-mulcom 11158 ax-addass 11159 ax-mulass 11160 ax-distr 11161 ax-i2m1 11162 ax-1ne0 11163 ax-1rid 11164 ax-rnegex 11165 ax-rrecex 11166 ax-cnre 11167 ax-pre-lttri 11168 ax-pre-lttrn 11169 ax-pre-ltadd 11170 ax-pre-mulgt0 11171 ax-pre-sup 11172 ax-addf 11173 ax-mulf 11174 ax-hilex 30179 ax-hfvadd 30180 ax-hvcom 30181 ax-hvass 30182 ax-hv0cl 30183 ax-hvaddid 30184 ax-hfvmul 30185 ax-hvmulid 30186 ax-hvmulass 30187 ax-hvdistr1 30188 ax-hvdistr2 30189 ax-hvmul0 30190 ax-hfi 30259 ax-his1 30262 ax-his2 30263 ax-his3 30264 ax-his4 30265 ax-hcompl 30382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-se 5626 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7350 df-ov 7397 df-oprab 7398 df-mpo 7399 df-of 7654 df-om 7840 df-1st 7959 df-2nd 7960 df-supp 8131 df-frecs 8250 df-wrecs 8281 df-recs 8355 df-rdg 8394 df-1o 8450 df-2o 8451 df-oadd 8454 df-omul 8455 df-er 8688 df-map 8807 df-pm 8808 df-ixp 8877 df-en 8925 df-dom 8926 df-sdom 8927 df-fin 8928 df-fsupp 9347 df-fi 9390 df-sup 9421 df-inf 9422 df-oi 9489 df-card 9918 df-acn 9921 df-pnf 11234 df-mnf 11235 df-xr 11236 df-ltxr 11237 df-le 11238 df-sub 11430 df-neg 11431 df-div 11856 df-nn 12197 df-2 12259 df-3 12260 df-4 12261 df-5 12262 df-6 12263 df-7 12264 df-8 12265 df-9 12266 df-n0 12457 df-z 12543 df-dec 12662 df-uz 12807 df-q 12917 df-rp 12959 df-xneg 13076 df-xadd 13077 df-xmul 13078 df-ioo 13312 df-ico 13314 df-icc 13315 df-fz 13469 df-fzo 13612 df-fl 13741 df-seq 13951 df-exp 14012 df-hash 14275 df-cj 15030 df-re 15031 df-im 15032 df-sqrt 15166 df-abs 15167 df-clim 15416 df-rlim 15417 df-sum 15617 df-struct 17064 df-sets 17081 df-slot 17099 df-ndx 17111 df-base 17129 df-ress 17158 df-plusg 17194 df-mulr 17195 df-starv 17196 df-sca 17197 df-vsca 17198 df-ip 17199 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-hom 17205 df-cco 17206 df-rest 17352 df-topn 17353 df-0g 17371 df-gsum 17372 df-topgen 17373 df-pt 17374 df-prds 17377 df-xrs 17432 df-qtop 17437 df-imas 17438 df-xps 17440 df-mre 17514 df-mrc 17515 df-acs 17517 df-mgm 18545 df-sgrp 18594 df-mnd 18605 df-submnd 18650 df-mulg 18925 df-cntz 19149 df-cmn 19616 df-psmet 20872 df-xmet 20873 df-met 20874 df-bl 20875 df-mopn 20876 df-fbas 20877 df-fg 20878 df-cnfld 20881 df-top 22327 df-topon 22344 df-topsp 22366 df-bases 22380 df-cld 22454 df-ntr 22455 df-cls 22456 df-nei 22533 df-cn 22662 df-cnp 22663 df-lm 22664 df-haus 22750 df-tx 22997 df-hmeo 23190 df-fil 23281 df-fm 23373 df-flim 23374 df-flf 23375 df-xms 23757 df-ms 23758 df-tms 23759 df-cfil 24703 df-cau 24704 df-cmet 24705 df-grpo 29673 df-gid 29674 df-ginv 29675 df-gdiv 29676 df-ablo 29725 df-vc 29739 df-nv 29772 df-va 29775 df-ba 29776 df-sm 29777 df-0v 29778 df-vs 29779 df-nmcv 29780 df-ims 29781 df-dip 29881 df-ssp 29902 df-ph 29993 df-cbn 30043 df-hnorm 30148 df-hba 30149 df-hvsub 30151 df-hlim 30152 df-hcau 30153 df-sh 30387 df-ch 30401 df-oc 30432 df-ch0 30433 df-shs 30488 df-pjh 30575 |
This theorem is referenced by: pjsubii 30858 pjaddi 30866 |
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