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Theorem ajfuni 30839

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 6516	. . 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 2731	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
3		eqid 2731	. . . . 5 ⊢ (·_𝑖OLD‘𝑈) = (·_𝑖OLD‘𝑈)
4		ajfuni.u	. . . . 5 ⊢ 𝑈 ∈ CPreHil_OLD
5	2, 3, 4	ajmoi 30838	. . . 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 2540	. . . 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 1800	. 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 30796	. . . 4 ⊢ 𝑈 ∈ NrmCVec
11		ajfuni.w	. . . 4 ⊢ 𝑊 ∈ NrmCVec
12		eqid 2731	. . . . 5 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
13		eqid 2731	. . . . 5 ⊢ (·_𝑖OLD‘𝑊) = (·_𝑖OLD‘𝑊)
14		ajfuni.5	. . . . 5 ⊢ 𝐴 = (𝑈adj𝑊)
15	2, 12, 3, 13, 14	ajfval 30789	. . . 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 692	. . 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 6502	. 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 1086 = wceq 1541 ∈ wcel 2111 ∃*wmo 2533 ∀wral 3047 {copab 5151 Fun wfun 6475 ⟶wf 6477 ‘cfv 6481 (class class class)co 7346 NrmCVeccnv 30564 BaseSetcba 30566 ·_𝑖OLDcdip 30680 adjcaj 30728 CPreHil_OLDccphlo 30792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-inf2 9531 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-pre-sup 11084 ax-addf 11085 ax-mulf 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-se 5568 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-isom 6490 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-supp 8091 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-ixp 8822 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-fsupp 9246 df-fi 9295 df-sup 9326 df-inf 9327 df-oi 9396 df-card 9832 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-div 11775 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-9 12195 df-n0 12382 df-z 12469 df-dec 12589 df-uz 12733 df-q 12847 df-rp 12891 df-xneg 13011 df-xadd 13012 df-xmul 13013 df-ioo 13249 df-icc 13252 df-fz 13408 df-fzo 13555 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-mulg 18981 df-cntz 19229 df-cmn 19694 df-psmet 21283 df-xmet 21284 df-met 21285 df-bl 21286 df-mopn 21287 df-cnfld 21292 df-top 22809 df-topon 22826 df-topsp 22848 df-bases 22861 df-cld 22934 df-ntr 22935 df-cls 22936 df-cn 23142 df-cnp 23143 df-t1 23229 df-haus 23230 df-tx 23477 df-hmeo 23670 df-xms 24235 df-ms 24236 df-tms 24237 df-grpo 30473 df-gid 30474 df-ginv 30475 df-gdiv 30476 df-ablo 30525 df-vc 30539 df-nv 30572 df-va 30575 df-ba 30576 df-sm 30577 df-0v 30578 df-vs 30579 df-nmcv 30580 df-ims 30581 df-dip 30681 df-aj 30730 df-ph 30793

This theorem is referenced by: ajfun 30840

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