ajfuni - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ajfuni		Structured version Visualization version GIF version

Theorem ajfuni 30893

Description: The adjoint function is a function. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ajfuni.5	⊢ 𝐴 = (𝑈adj𝑊)
ajfuni.u	⊢ 𝑈 ∈ CPreHil_OLD
ajfuni.w	⊢ 𝑊 ∈ NrmCVec

**Assertion**
Ref	Expression
ajfuni	⊢ Fun 𝐴

Proof of Theorem ajfuni Dummy variables 𝑡 𝑠 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funopab 6615	. . 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 2740	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
3		eqid 2740	. . . . 5 ⊢ (·_𝑖OLD‘𝑈) = (·_𝑖OLD‘𝑈)
4		ajfuni.u	. . . . 5 ⊢ 𝑈 ∈ CPreHil_OLD
5	2, 3, 4	ajmoi 30892	. . . 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 1150	. . . . 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 2548	. . . 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 1797	. 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 30850	. . . 4 ⊢ 𝑈 ∈ NrmCVec
11		ajfuni.w	. . . 4 ⊢ 𝑊 ∈ NrmCVec
12		eqid 2740	. . . . 5 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
13		eqid 2740	. . . . 5 ⊢ (·_𝑖OLD‘𝑊) = (·_𝑖OLD‘𝑊)
14		ajfuni.5	. . . . 5 ⊢ 𝐴 = (𝑈adj𝑊)
15	2, 12, 3, 13, 14	ajfval 30843	. . . 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 691	. . 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 6601	. 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 231	1 ⊢ Fun 𝐴

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∃*wmo 2541 ∀wral 3067 {copab 5228 Fun wfun 6569 ⟶wf 6571 ‘cfv 6575 (class class class)co 7450 NrmCVeccnv 30618 BaseSetcba 30620 ·_𝑖OLDcdip 30734 adjcaj 30782 CPreHil_OLDccphlo 30846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 ax-addf 11265 ax-mulf 11266

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-om 7906 df-1st 8032 df-2nd 8033 df-supp 8204 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-2o 8525 df-er 8765 df-map 8888 df-ixp 8958 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-fsupp 9434 df-fi 9482 df-sup 9513 df-inf 9514 df-oi 9581 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-7 12363 df-8 12364 df-9 12365 df-n0 12556 df-z 12642 df-dec 12761 df-uz 12906 df-q 13016 df-rp 13060 df-xneg 13177 df-xadd 13178 df-xmul 13179 df-ioo 13413 df-icc 13416 df-fz 13570 df-fzo 13714 df-seq 14055 df-exp 14115 df-hash 14382 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15536 df-sum 15737 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-starv 17328 df-sca 17329 df-vsca 17330 df-ip 17331 df-tset 17332 df-ple 17333 df-ds 17335 df-unif 17336 df-hom 17337 df-cco 17338 df-rest 17484 df-topn 17485 df-0g 17503 df-gsum 17504 df-topgen 17505 df-pt 17506 df-prds 17509 df-xrs 17564 df-qtop 17569 df-imas 17570 df-xps 17572 df-mre 17646 df-mrc 17647 df-acs 17649 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-submnd 18821 df-mulg 19110 df-cntz 19359 df-cmn 19826 df-psmet 21381 df-xmet 21382 df-met 21383 df-bl 21384 df-mopn 21385 df-cnfld 21390 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22976 df-cld 23050 df-ntr 23051 df-cls 23052 df-cn 23258 df-cnp 23259 df-t1 23345 df-haus 23346 df-tx 23593 df-hmeo 23786 df-xms 24353 df-ms 24354 df-tms 24355 df-grpo 30527 df-gid 30528 df-ginv 30529 df-gdiv 30530 df-ablo 30579 df-vc 30593 df-nv 30626 df-va 30629 df-ba 30630 df-sm 30631 df-0v 30632 df-vs 30633 df-nmcv 30634 df-ims 30635 df-dip 30735 df-aj 30784 df-ph 30847

This theorem is referenced by: ajfun 30894

W3C validator