Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > ajfuni | Structured version Visualization version GIF version |
Description: The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ajfuni.5 | ⊢ 𝐴 = (𝑈adj𝑊) |
ajfuni.u | ⊢ 𝑈 ∈ CPreHilOLD |
ajfuni.w | ⊢ 𝑊 ∈ NrmCVec |
Ref | Expression |
---|---|
ajfuni | ⊢ Fun 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funopab 5923 | . . 3 ⊢ (Fun {〈𝑡, 𝑠〉 ∣ (𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦)))} ↔ ∀𝑡∃*𝑠(𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦)))) | |
2 | eqid 2622 | . . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈) | |
3 | eqid 2622 | . . . . 5 ⊢ (·𝑖OLD‘𝑈) = (·𝑖OLD‘𝑈) | |
4 | ajfuni.u | . . . . 5 ⊢ 𝑈 ∈ CPreHilOLD | |
5 | 2, 3, 4 | ajmoi 27714 | . . . 4 ⊢ ∃*𝑠(𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦))) |
6 | 3simpc 1060 | . . . . 5 ⊢ ((𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦))) → (𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦)))) | |
7 | 6 | moimi 2520 | . . . 4 ⊢ (∃*𝑠(𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦))) → ∃*𝑠(𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦)))) |
8 | 5, 7 | ax-mp 5 | . . 3 ⊢ ∃*𝑠(𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦))) |
9 | 1, 8 | mpgbir 1726 | . 2 ⊢ Fun {〈𝑡, 𝑠〉 ∣ (𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦)))} |
10 | 4 | phnvi 27671 | . . . 4 ⊢ 𝑈 ∈ NrmCVec |
11 | ajfuni.w | . . . 4 ⊢ 𝑊 ∈ NrmCVec | |
12 | eqid 2622 | . . . . 5 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊) | |
13 | eqid 2622 | . . . . 5 ⊢ (·𝑖OLD‘𝑊) = (·𝑖OLD‘𝑊) | |
14 | ajfuni.5 | . . . . 5 ⊢ 𝐴 = (𝑈adj𝑊) | |
15 | 2, 12, 3, 13, 14 | ajfval 27664 | . . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ NrmCVec) → 𝐴 = {〈𝑡, 𝑠〉 ∣ (𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦)))}) |
16 | 10, 11, 15 | mp2an 708 | . . 3 ⊢ 𝐴 = {〈𝑡, 𝑠〉 ∣ (𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦)))} |
17 | 16 | funeqi 5909 | . 2 ⊢ (Fun 𝐴 ↔ Fun {〈𝑡, 𝑠〉 ∣ (𝑡:(BaseSet‘𝑈)⟶(BaseSet‘𝑊) ∧ 𝑠:(BaseSet‘𝑊)⟶(BaseSet‘𝑈) ∧ ∀𝑥 ∈ (BaseSet‘𝑈)∀𝑦 ∈ (BaseSet‘𝑊)((𝑡‘𝑥)(·𝑖OLD‘𝑊)𝑦) = (𝑥(·𝑖OLD‘𝑈)(𝑠‘𝑦)))}) |
18 | 9, 17 | mpbir 221 | 1 ⊢ Fun 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∃*wmo 2471 ∀wral 2912 {copab 4712 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 NrmCVeccnv 27439 BaseSetcba 27441 ·𝑖OLDcdip 27555 adjcaj 27603 CPreHilOLDccphlo 27667 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-cn 21031 df-cnp 21032 df-t1 21118 df-haus 21119 df-tx 21365 df-hmeo 21558 df-xms 22125 df-ms 22126 df-tms 22127 df-grpo 27347 df-gid 27348 df-ginv 27349 df-gdiv 27350 df-ablo 27399 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-vs 27454 df-nmcv 27455 df-ims 27456 df-dip 27556 df-aj 27605 df-ph 27668 |
This theorem is referenced by: ajfun 27716 |
Copyright terms: Public domain | W3C validator |