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Mirrors > Home > MPE Home > Th. List > ajfun | Structured version Visualization version GIF version |
Description: The adjoint function is a function. This is not immediately apparent from df-aj 28677 but results from the uniqueness shown by ajmoi 28785. (Contributed by NM, 26-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ajfun.5 | ⊢ 𝐴 = (𝑈adj𝑊) |
Ref | Expression |
---|---|
ajfun | ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec) → Fun 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ajfun.5 | . . . 4 ⊢ 𝐴 = (𝑈adj𝑊) | |
2 | oveq1 7171 | . . . 4 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (𝑈adj𝑊) = (if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)adj𝑊)) | |
3 | 1, 2 | syl5eq 2785 | . . 3 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → 𝐴 = (if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)adj𝑊)) |
4 | 3 | funeqd 6355 | . 2 ⊢ (𝑈 = if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) → (Fun 𝐴 ↔ Fun (if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)adj𝑊))) |
5 | oveq2 7172 | . . 3 ⊢ (𝑊 = if(𝑊 ∈ NrmCVec, 𝑊, 〈〈 + , · 〉, abs〉) → (if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)adj𝑊) = (if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)adjif(𝑊 ∈ NrmCVec, 𝑊, 〈〈 + , · 〉, abs〉))) | |
6 | 5 | funeqd 6355 | . 2 ⊢ (𝑊 = if(𝑊 ∈ NrmCVec, 𝑊, 〈〈 + , · 〉, abs〉) → (Fun (if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)adj𝑊) ↔ Fun (if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)adjif(𝑊 ∈ NrmCVec, 𝑊, 〈〈 + , · 〉, abs〉)))) |
7 | eqid 2738 | . . 3 ⊢ (if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)adjif(𝑊 ∈ NrmCVec, 𝑊, 〈〈 + , · 〉, abs〉)) = (if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)adjif(𝑊 ∈ NrmCVec, 𝑊, 〈〈 + , · 〉, abs〉)) | |
8 | elimphu 28748 | . . 3 ⊢ if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉) ∈ CPreHilOLD | |
9 | elimnvu 28611 | . . 3 ⊢ if(𝑊 ∈ NrmCVec, 𝑊, 〈〈 + , · 〉, abs〉) ∈ NrmCVec | |
10 | 7, 8, 9 | ajfuni 28786 | . 2 ⊢ Fun (if(𝑈 ∈ CPreHilOLD, 𝑈, 〈〈 + , · 〉, abs〉)adjif(𝑊 ∈ NrmCVec, 𝑊, 〈〈 + , · 〉, abs〉)) |
11 | 4, 6, 10 | dedth2h 4470 | 1 ⊢ ((𝑈 ∈ CPreHilOLD ∧ 𝑊 ∈ NrmCVec) → Fun 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2113 ifcif 4411 〈cop 4519 Fun wfun 6327 (class class class)co 7164 + caddc 10611 · cmul 10613 abscabs 14676 NrmCVeccnv 28511 adjcaj 28675 CPreHilOLDccphlo 28739 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-pre-sup 10686 ax-addf 10687 ax-mulf 10688 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-isom 6342 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-of 7419 df-om 7594 df-1st 7707 df-2nd 7708 df-supp 7850 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-2o 8125 df-er 8313 df-map 8432 df-ixp 8501 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-fsupp 8900 df-fi 8941 df-sup 8972 df-inf 8973 df-oi 9040 df-card 9434 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-div 11369 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-7 11777 df-8 11778 df-9 11779 df-n0 11970 df-z 12056 df-dec 12173 df-uz 12318 df-q 12424 df-rp 12466 df-xneg 12583 df-xadd 12584 df-xmul 12585 df-ioo 12818 df-icc 12821 df-fz 12975 df-fzo 13118 df-seq 13454 df-exp 13515 df-hash 13776 df-cj 14541 df-re 14542 df-im 14543 df-sqrt 14677 df-abs 14678 df-clim 14928 df-sum 15129 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-starv 16676 df-sca 16677 df-vsca 16678 df-ip 16679 df-tset 16680 df-ple 16681 df-ds 16683 df-unif 16684 df-hom 16685 df-cco 16686 df-rest 16792 df-topn 16793 df-0g 16811 df-gsum 16812 df-topgen 16813 df-pt 16814 df-prds 16817 df-xrs 16871 df-qtop 16876 df-imas 16877 df-xps 16879 df-mre 16953 df-mrc 16954 df-acs 16956 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-mulg 18336 df-cntz 18558 df-cmn 19019 df-psmet 20202 df-xmet 20203 df-met 20204 df-bl 20205 df-mopn 20206 df-cnfld 20211 df-top 21638 df-topon 21655 df-topsp 21677 df-bases 21690 df-cld 21763 df-ntr 21764 df-cls 21765 df-cn 21971 df-cnp 21972 df-t1 22058 df-haus 22059 df-tx 22306 df-hmeo 22499 df-xms 23066 df-ms 23067 df-tms 23068 df-grpo 28420 df-gid 28421 df-ginv 28422 df-gdiv 28423 df-ablo 28472 df-vc 28486 df-nv 28519 df-va 28522 df-ba 28523 df-sm 28524 df-0v 28525 df-vs 28526 df-nmcv 28527 df-ims 28528 df-dip 28628 df-aj 28677 df-ph 28740 |
This theorem is referenced by: (None) |
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