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Mirrors > Home > MPE Home > Th. List > ajmoi | Structured version Visualization version GIF version |
Description: Every operator has at most one adjoint. (Contributed by NM, 25-Jan-2008.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ip2eqi.1 | ⊢ 𝑋 = (BaseSet‘𝑈) |
ip2eqi.7 | ⊢ 𝑃 = (·𝑖OLD‘𝑈) |
ip2eqi.u | ⊢ 𝑈 ∈ CPreHilOLD |
Ref | Expression |
---|---|
ajmoi | ⊢ ∃*𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.26-2 3168 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))) ↔ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) | |
2 | eqtr2 2839 | . . . . . . 7 ⊢ ((((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))) → (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦))) | |
3 | 2 | 2ralimi 3158 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦))) |
4 | 1, 3 | sylbir 236 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))) → ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦))) |
5 | ip2eqi.1 | . . . . . . 7 ⊢ 𝑋 = (BaseSet‘𝑈) | |
6 | ip2eqi.7 | . . . . . . 7 ⊢ 𝑃 = (·𝑖OLD‘𝑈) | |
7 | ip2eqi.u | . . . . . . 7 ⊢ 𝑈 ∈ CPreHilOLD | |
8 | 5, 6, 7 | phoeqi 28561 | . . . . . 6 ⊢ ((𝑠:𝑌⟶𝑋 ∧ 𝑡:𝑌⟶𝑋) → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦)) ↔ 𝑠 = 𝑡)) |
9 | 8 | biimpa 477 | . . . . 5 ⊢ (((𝑠:𝑌⟶𝑋 ∧ 𝑡:𝑌⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦))) → 𝑠 = 𝑡) |
10 | 4, 9 | sylan2 592 | . . . 4 ⊢ (((𝑠:𝑌⟶𝑋 ∧ 𝑡:𝑌⟶𝑋) ∧ (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) → 𝑠 = 𝑡) |
11 | 10 | an4s 656 | . . 3 ⊢ (((𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ∧ (𝑡:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) → 𝑠 = 𝑡) |
12 | 11 | gen2 1788 | . 2 ⊢ ∀𝑠∀𝑡(((𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ∧ (𝑡:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) → 𝑠 = 𝑡) |
13 | feq1 6488 | . . . 4 ⊢ (𝑠 = 𝑡 → (𝑠:𝑌⟶𝑋 ↔ 𝑡:𝑌⟶𝑋)) | |
14 | fveq1 6662 | . . . . . . 7 ⊢ (𝑠 = 𝑡 → (𝑠‘𝑦) = (𝑡‘𝑦)) | |
15 | 14 | oveq2d 7161 | . . . . . 6 ⊢ (𝑠 = 𝑡 → (𝑥𝑃(𝑠‘𝑦)) = (𝑥𝑃(𝑡‘𝑦))) |
16 | 15 | eqeq2d 2829 | . . . . 5 ⊢ (𝑠 = 𝑡 → (((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) |
17 | 16 | 2ralbidv 3196 | . . . 4 ⊢ (𝑠 = 𝑡 → (∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦)) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) |
18 | 13, 17 | anbi12d 630 | . . 3 ⊢ (𝑠 = 𝑡 → ((𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ (𝑡:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦))))) |
19 | 18 | mo4 2643 | . 2 ⊢ (∃*𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ↔ ∀𝑠∀𝑡(((𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) ∧ (𝑡:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑡‘𝑦)))) → 𝑠 = 𝑡)) |
20 | 12, 19 | mpbir 232 | 1 ⊢ ∃*𝑠(𝑠:𝑌⟶𝑋 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑌 ((𝑇‘𝑥)𝑄𝑦) = (𝑥𝑃(𝑠‘𝑦))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1526 = wceq 1528 ∈ wcel 2105 ∃*wmo 2613 ∀wral 3135 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 BaseSetcba 28290 ·𝑖OLDcdip 28404 CPreHilOLDccphlo 28516 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-mulg 18163 df-cntz 18385 df-cmn 18837 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-cnfld 20474 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-cn 21763 df-cnp 21764 df-t1 21850 df-haus 21851 df-tx 22098 df-hmeo 22291 df-xms 22857 df-ms 22858 df-tms 22859 df-grpo 28197 df-gid 28198 df-ginv 28199 df-gdiv 28200 df-ablo 28249 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-vs 28303 df-nmcv 28304 df-ims 28305 df-dip 28405 df-ph 28517 |
This theorem is referenced by: ajfuni 28563 |
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