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Mirrors > Home > HSE Home > Th. List > pj3i | Structured version Visualization version GIF version |
Description: Projection triplet theorem. (Contributed by NM, 2-Dec-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjadj2co.1 | ⊢ 𝐹 ∈ Cℋ |
pjadj2co.2 | ⊢ 𝐺 ∈ Cℋ |
pjadj2co.3 | ⊢ 𝐻 ∈ Cℋ |
Ref | Expression |
---|---|
pj3i | ⊢ (((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐹)) ∧ (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐺) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐻))) → (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (projℎ‘((𝐹 ∩ 𝐺) ∩ 𝐻))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coass 6264 | . . . . 5 ⊢ (((projℎ‘𝐺) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐺) ∘ ((projℎ‘𝐹) ∘ (projℎ‘𝐻))) | |
2 | eqeq1 2732 | . . . . 5 ⊢ ((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐺) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐻)) → ((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐺) ∘ ((projℎ‘𝐹) ∘ (projℎ‘𝐻))) ↔ (((projℎ‘𝐺) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐺) ∘ ((projℎ‘𝐹) ∘ (projℎ‘𝐻))))) | |
3 | 1, 2 | mpbiri 258 | . . . 4 ⊢ ((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐺) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐻)) → (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = ((projℎ‘𝐺) ∘ ((projℎ‘𝐹) ∘ (projℎ‘𝐻)))) |
4 | 3 | rneqd 5935 | . . 3 ⊢ ((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐺) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐻)) → ran (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = ran ((projℎ‘𝐺) ∘ ((projℎ‘𝐹) ∘ (projℎ‘𝐻)))) |
5 | rncoss 5970 | . . . 4 ⊢ ran ((projℎ‘𝐺) ∘ ((projℎ‘𝐹) ∘ (projℎ‘𝐻))) ⊆ ran (projℎ‘𝐺) | |
6 | pjadj2co.2 | . . . . 5 ⊢ 𝐺 ∈ Cℋ | |
7 | 6 | pjrni 31506 | . . . 4 ⊢ ran (projℎ‘𝐺) = 𝐺 |
8 | 5, 7 | sseqtri 4015 | . . 3 ⊢ ran ((projℎ‘𝐺) ∘ ((projℎ‘𝐹) ∘ (projℎ‘𝐻))) ⊆ 𝐺 |
9 | 4, 8 | eqsstrdi 4033 | . 2 ⊢ ((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐺) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐻)) → ran (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) ⊆ 𝐺) |
10 | pjadj2co.1 | . . 3 ⊢ 𝐹 ∈ Cℋ | |
11 | pjadj2co.3 | . . 3 ⊢ 𝐻 ∈ Cℋ | |
12 | 10, 6, 11 | pj3si 32011 | . 2 ⊢ (((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐹)) ∧ ran (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) ⊆ 𝐺) → (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (projℎ‘((𝐹 ∩ 𝐺) ∩ 𝐻))) |
13 | 9, 12 | sylan2 592 | 1 ⊢ (((((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐻) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐹)) ∧ (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (((projℎ‘𝐺) ∘ (projℎ‘𝐹)) ∘ (projℎ‘𝐻))) → (((projℎ‘𝐹) ∘ (projℎ‘𝐺)) ∘ (projℎ‘𝐻)) = (projℎ‘((𝐹 ∩ 𝐺) ∩ 𝐻))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3944 ⊆ wss 3945 ran crn 5674 ∘ ccom 5677 ‘cfv 6543 Cℋ cch 30733 projℎcpjh 30741 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-inf2 9659 ax-cc 10453 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-pre-sup 11211 ax-addf 11212 ax-mulf 11213 ax-hilex 30803 ax-hfvadd 30804 ax-hvcom 30805 ax-hvass 30806 ax-hv0cl 30807 ax-hvaddid 30808 ax-hfvmul 30809 ax-hvmulid 30810 ax-hvmulass 30811 ax-hvdistr1 30812 ax-hvdistr2 30813 ax-hvmul0 30814 ax-hfi 30883 ax-his1 30886 ax-his2 30887 ax-his3 30888 ax-his4 30889 ax-hcompl 31006 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-se 5629 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-supp 8161 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-2o 8482 df-oadd 8485 df-omul 8486 df-er 8719 df-map 8841 df-pm 8842 df-ixp 8911 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-fsupp 9381 df-fi 9429 df-sup 9460 df-inf 9461 df-oi 9528 df-card 9957 df-acn 9960 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-div 11897 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-7 12305 df-8 12306 df-9 12307 df-n0 12498 df-z 12584 df-dec 12703 df-uz 12848 df-q 12958 df-rp 13002 df-xneg 13119 df-xadd 13120 df-xmul 13121 df-ioo 13355 df-ico 13357 df-icc 13358 df-fz 13512 df-fzo 13655 df-fl 13784 df-seq 13994 df-exp 14054 df-hash 14317 df-cj 15073 df-re 15074 df-im 15075 df-sqrt 15209 df-abs 15210 df-clim 15459 df-rlim 15460 df-sum 15660 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17398 df-topn 17399 df-0g 17417 df-gsum 17418 df-topgen 17419 df-pt 17420 df-prds 17423 df-xrs 17478 df-qtop 17483 df-imas 17484 df-xps 17486 df-mre 17560 df-mrc 17561 df-acs 17563 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-mulg 19018 df-cntz 19262 df-cmn 19731 df-psmet 21265 df-xmet 21266 df-met 21267 df-bl 21268 df-mopn 21269 df-fbas 21270 df-fg 21271 df-cnfld 21274 df-top 22790 df-topon 22807 df-topsp 22829 df-bases 22843 df-cld 22917 df-ntr 22918 df-cls 22919 df-nei 22996 df-cn 23125 df-cnp 23126 df-lm 23127 df-haus 23213 df-tx 23460 df-hmeo 23653 df-fil 23744 df-fm 23836 df-flim 23837 df-flf 23838 df-xms 24220 df-ms 24221 df-tms 24222 df-cfil 25177 df-cau 25178 df-cmet 25179 df-grpo 30297 df-gid 30298 df-ginv 30299 df-gdiv 30300 df-ablo 30349 df-vc 30363 df-nv 30396 df-va 30399 df-ba 30400 df-sm 30401 df-0v 30402 df-vs 30403 df-nmcv 30404 df-ims 30405 df-dip 30505 df-ssp 30526 df-ph 30617 df-cbn 30667 df-hnorm 30772 df-hba 30773 df-hvsub 30775 df-hlim 30776 df-hcau 30777 df-sh 31011 df-ch 31025 df-oc 31056 df-ch0 31057 df-shs 31112 df-pjh 31199 |
This theorem is referenced by: pj3cor1i 32013 |
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