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Theorem ajfuni 30822

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 6521	. . 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 2729	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
3		eqid 2729	. . . . 5 ⊢ (·_𝑖OLD‘𝑈) = (·_𝑖OLD‘𝑈)
4		ajfuni.u	. . . . 5 ⊢ 𝑈 ∈ CPreHil_OLD
5	2, 3, 4	ajmoi 30821	. . . 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 2538	. . . 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 1799	. 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 30779	. . . 4 ⊢ 𝑈 ∈ NrmCVec
11		ajfuni.w	. . . 4 ⊢ 𝑊 ∈ NrmCVec
12		eqid 2729	. . . . 5 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
13		eqid 2729	. . . . 5 ⊢ (·_𝑖OLD‘𝑊) = (·_𝑖OLD‘𝑊)
14		ajfuni.5	. . . . 5 ⊢ 𝐴 = (𝑈adj𝑊)
15	2, 12, 3, 13, 14	ajfval 30772	. . . 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 6507	. 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 1540 ∈ wcel 2109 ∃*wmo 2531 ∀wral 3044 {copab 5157 Fun wfun 6480 ⟶wf 6482 ‘cfv 6486 (class class class)co 7353 NrmCVeccnv 30547 BaseSetcba 30549 ·_𝑖OLDcdip 30663 adjcaj 30711 CPreHil_OLDccphlo 30775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7675 ax-inf2 9556 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107 ax-mulf 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3345 df-reu 3346 df-rab 3397 df-v 3440 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4479 df-pw 4555 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4862 df-int 4900 df-iun 4946 df-iin 4947 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5518 df-eprel 5523 df-po 5531 df-so 5532 df-fr 5576 df-se 5577 df-we 5578 df-xp 5629 df-rel 5630 df-cnv 5631 df-co 5632 df-dm 5633 df-rn 5634 df-res 5635 df-ima 5636 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-isom 6495 df-riota 7310 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7617 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8632 df-map 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9271 df-fi 9320 df-sup 9351 df-inf 9352 df-oi 9421 df-card 9854 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12611 df-uz 12755 df-q 12869 df-rp 12913 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-ioo 13271 df-icc 13274 df-fz 13430 df-fzo 13577 df-seq 13928 df-exp 13988 df-hash 14257 df-cj 15025 df-re 15026 df-im 15027 df-sqrt 15161 df-abs 15162 df-clim 15414 df-sum 15613 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17140 df-ress 17161 df-plusg 17193 df-mulr 17194 df-starv 17195 df-sca 17196 df-vsca 17197 df-ip 17198 df-tset 17199 df-ple 17200 df-ds 17202 df-unif 17203 df-hom 17204 df-cco 17205 df-rest 17345 df-topn 17346 df-0g 17364 df-gsum 17365 df-topgen 17366 df-pt 17367 df-prds 17370 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18533 df-sgrp 18612 df-mnd 18628 df-submnd 18677 df-mulg 18966 df-cntz 19215 df-cmn 19680 df-psmet 21272 df-xmet 21273 df-met 21274 df-bl 21275 df-mopn 21276 df-cnfld 21281 df-top 22798 df-topon 22815 df-topsp 22837 df-bases 22850 df-cld 22923 df-ntr 22924 df-cls 22925 df-cn 23131 df-cnp 23132 df-t1 23218 df-haus 23219 df-tx 23466 df-hmeo 23659 df-xms 24225 df-ms 24226 df-tms 24227 df-grpo 30456 df-gid 30457 df-ginv 30458 df-gdiv 30459 df-ablo 30508 df-vc 30522 df-nv 30555 df-va 30558 df-ba 30559 df-sm 30560 df-0v 30561 df-vs 30562 df-nmcv 30563 df-ims 30564 df-dip 30664 df-aj 30713 df-ph 30776

This theorem is referenced by: ajfun 30823

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