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Theorem ajfuni 30786

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 6570	. . 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 2735	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
3		eqid 2735	. . . . 5 ⊢ (·_𝑖OLD‘𝑈) = (·_𝑖OLD‘𝑈)
4		ajfuni.u	. . . . 5 ⊢ 𝑈 ∈ CPreHil_OLD
5	2, 3, 4	ajmoi 30785	. . . 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 2544	. . . 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 30743	. . . 4 ⊢ 𝑈 ∈ NrmCVec
11		ajfuni.w	. . . 4 ⊢ 𝑊 ∈ NrmCVec
12		eqid 2735	. . . . 5 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
13		eqid 2735	. . . . 5 ⊢ (·_𝑖OLD‘𝑊) = (·_𝑖OLD‘𝑊)
14		ajfuni.5	. . . . 5 ⊢ 𝐴 = (𝑈adj𝑊)
15	2, 12, 3, 13, 14	ajfval 30736	. . . 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 6556	. 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 2108 ∃*wmo 2537 ∀wral 3051 {copab 5181 Fun wfun 6524 ⟶wf 6526 ‘cfv 6530 (class class class)co 7403 NrmCVeccnv 30511 BaseSetcba 30513 ·_𝑖OLDcdip 30627 adjcaj 30675 CPreHil_OLDccphlo 30739

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7727 ax-inf2 9653 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204 ax-pre-sup 11205 ax-addf 11206 ax-mulf 11207

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6483 df-fun 6532 df-fn 6533 df-f 6534 df-f1 6535 df-fo 6536 df-f1o 6537 df-fv 6538 df-isom 6539 df-riota 7360 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7669 df-om 7860 df-1st 7986 df-2nd 7987 df-supp 8158 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-er 8717 df-map 8840 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9372 df-fi 9421 df-sup 9452 df-inf 9453 df-oi 9522 df-card 9951 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11466 df-neg 11467 df-div 11893 df-nn 12239 df-2 12301 df-3 12302 df-4 12303 df-5 12304 df-6 12305 df-7 12306 df-8 12307 df-9 12308 df-n0 12500 df-z 12587 df-dec 12707 df-uz 12851 df-q 12963 df-rp 13007 df-xneg 13126 df-xadd 13127 df-xmul 13128 df-ioo 13364 df-icc 13367 df-fz 13523 df-fzo 13670 df-seq 14018 df-exp 14078 df-hash 14347 df-cj 15116 df-re 15117 df-im 15118 df-sqrt 15252 df-abs 15253 df-clim 15502 df-sum 15701 df-struct 17164 df-sets 17181 df-slot 17199 df-ndx 17211 df-base 17227 df-ress 17250 df-plusg 17282 df-mulr 17283 df-starv 17284 df-sca 17285 df-vsca 17286 df-ip 17287 df-tset 17288 df-ple 17289 df-ds 17291 df-unif 17292 df-hom 17293 df-cco 17294 df-rest 17434 df-topn 17435 df-0g 17453 df-gsum 17454 df-topgen 17455 df-pt 17456 df-prds 17459 df-xrs 17514 df-qtop 17519 df-imas 17520 df-xps 17522 df-mre 17596 df-mrc 17597 df-acs 17599 df-mgm 18616 df-sgrp 18695 df-mnd 18711 df-submnd 18760 df-mulg 19049 df-cntz 19298 df-cmn 19761 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-cnfld 21314 df-top 22830 df-topon 22847 df-topsp 22869 df-bases 22882 df-cld 22955 df-ntr 22956 df-cls 22957 df-cn 23163 df-cnp 23164 df-t1 23250 df-haus 23251 df-tx 23498 df-hmeo 23691 df-xms 24257 df-ms 24258 df-tms 24259 df-grpo 30420 df-gid 30421 df-ginv 30422 df-gdiv 30423 df-ablo 30472 df-vc 30486 df-nv 30519 df-va 30522 df-ba 30523 df-sm 30524 df-0v 30525 df-vs 30526 df-nmcv 30527 df-ims 30528 df-dip 30628 df-aj 30677 df-ph 30740

This theorem is referenced by: ajfun 30787

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