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Mirrors > Home > HSE Home > Th. List > pjhtheu2 | Structured version Visualization version GIF version |
Description: Uniqueness of 𝑦 for the projection theorem. (Contributed by NM, 6-Nov-1999.) (Revised by Mario Carneiro, 15-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
pjhtheu2 | ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃!𝑦 ∈ (⊥‘𝐻)∃𝑥 ∈ 𝐻 𝐴 = (𝑥 +ℎ 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | choccl 28737 | . . 3 ⊢ (𝐻 ∈ Cℋ → (⊥‘𝐻) ∈ Cℋ ) | |
2 | pjhtheu 28825 | . . 3 ⊢ (((⊥‘𝐻) ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃!𝑦 ∈ (⊥‘𝐻)∃𝑥 ∈ (⊥‘(⊥‘𝐻))𝐴 = (𝑦 +ℎ 𝑥)) | |
3 | 1, 2 | sylan 575 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃!𝑦 ∈ (⊥‘𝐻)∃𝑥 ∈ (⊥‘(⊥‘𝐻))𝐴 = (𝑦 +ℎ 𝑥)) |
4 | simpll 757 | . . . . . 6 ⊢ (((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝑦 ∈ (⊥‘𝐻)) → 𝐻 ∈ Cℋ ) | |
5 | ococ 28837 | . . . . . 6 ⊢ (𝐻 ∈ Cℋ → (⊥‘(⊥‘𝐻)) = 𝐻) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝑦 ∈ (⊥‘𝐻)) → (⊥‘(⊥‘𝐻)) = 𝐻) |
7 | 6 | rexeqdv 3340 | . . . 4 ⊢ (((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝑦 ∈ (⊥‘𝐻)) → (∃𝑥 ∈ (⊥‘(⊥‘𝐻))𝐴 = (𝑦 +ℎ 𝑥) ↔ ∃𝑥 ∈ 𝐻 𝐴 = (𝑦 +ℎ 𝑥))) |
8 | 1 | adantr 474 | . . . . . . . . 9 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (⊥‘𝐻) ∈ Cℋ ) |
9 | chel 28659 | . . . . . . . . 9 ⊢ (((⊥‘𝐻) ∈ Cℋ ∧ 𝑦 ∈ (⊥‘𝐻)) → 𝑦 ∈ ℋ) | |
10 | 8, 9 | sylan 575 | . . . . . . . 8 ⊢ (((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝑦 ∈ (⊥‘𝐻)) → 𝑦 ∈ ℋ) |
11 | 10 | adantr 474 | . . . . . . 7 ⊢ ((((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝑦 ∈ (⊥‘𝐻)) ∧ 𝑥 ∈ 𝐻) → 𝑦 ∈ ℋ) |
12 | chel 28659 | . . . . . . . 8 ⊢ ((𝐻 ∈ Cℋ ∧ 𝑥 ∈ 𝐻) → 𝑥 ∈ ℋ) | |
13 | 4, 12 | sylan 575 | . . . . . . 7 ⊢ ((((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝑦 ∈ (⊥‘𝐻)) ∧ 𝑥 ∈ 𝐻) → 𝑥 ∈ ℋ) |
14 | ax-hvcom 28430 | . . . . . . 7 ⊢ ((𝑦 ∈ ℋ ∧ 𝑥 ∈ ℋ) → (𝑦 +ℎ 𝑥) = (𝑥 +ℎ 𝑦)) | |
15 | 11, 13, 14 | syl2anc 579 | . . . . . 6 ⊢ ((((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝑦 ∈ (⊥‘𝐻)) ∧ 𝑥 ∈ 𝐻) → (𝑦 +ℎ 𝑥) = (𝑥 +ℎ 𝑦)) |
16 | 15 | eqeq2d 2787 | . . . . 5 ⊢ ((((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝑦 ∈ (⊥‘𝐻)) ∧ 𝑥 ∈ 𝐻) → (𝐴 = (𝑦 +ℎ 𝑥) ↔ 𝐴 = (𝑥 +ℎ 𝑦))) |
17 | 16 | rexbidva 3233 | . . . 4 ⊢ (((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝑦 ∈ (⊥‘𝐻)) → (∃𝑥 ∈ 𝐻 𝐴 = (𝑦 +ℎ 𝑥) ↔ ∃𝑥 ∈ 𝐻 𝐴 = (𝑥 +ℎ 𝑦))) |
18 | 7, 17 | bitrd 271 | . . 3 ⊢ (((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) ∧ 𝑦 ∈ (⊥‘𝐻)) → (∃𝑥 ∈ (⊥‘(⊥‘𝐻))𝐴 = (𝑦 +ℎ 𝑥) ↔ ∃𝑥 ∈ 𝐻 𝐴 = (𝑥 +ℎ 𝑦))) |
19 | 18 | reubidva 3311 | . 2 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → (∃!𝑦 ∈ (⊥‘𝐻)∃𝑥 ∈ (⊥‘(⊥‘𝐻))𝐴 = (𝑦 +ℎ 𝑥) ↔ ∃!𝑦 ∈ (⊥‘𝐻)∃𝑥 ∈ 𝐻 𝐴 = (𝑥 +ℎ 𝑦))) |
20 | 3, 19 | mpbid 224 | 1 ⊢ ((𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ) → ∃!𝑦 ∈ (⊥‘𝐻)∃𝑥 ∈ 𝐻 𝐴 = (𝑥 +ℎ 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 ∃wrex 3090 ∃!wreu 3091 ‘cfv 6135 (class class class)co 6922 ℋchba 28348 +ℎ cva 28349 Cℋ cch 28358 ⊥cort 28359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cc 9592 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 ax-hilex 28428 ax-hfvadd 28429 ax-hvcom 28430 ax-hvass 28431 ax-hv0cl 28432 ax-hvaddid 28433 ax-hfvmul 28434 ax-hvmulid 28435 ax-hvmulass 28436 ax-hvdistr1 28437 ax-hvdistr2 28438 ax-hvmul0 28439 ax-hfi 28508 ax-his1 28511 ax-his2 28512 ax-his3 28513 ax-his4 28514 ax-hcompl 28631 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-iin 4756 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-supp 7577 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-omul 7848 df-er 8026 df-map 8142 df-pm 8143 df-ixp 8195 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-fsupp 8564 df-fi 8605 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-acn 9101 df-cda 9325 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-ioo 12491 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-rlim 14628 df-sum 14825 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-ip 16356 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-hom 16362 df-cco 16363 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-pt 16491 df-prds 16494 df-xrs 16548 df-qtop 16553 df-imas 16554 df-xps 16556 df-mre 16632 df-mrc 16633 df-acs 16635 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-submnd 17722 df-mulg 17928 df-cntz 18133 df-cmn 18581 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-cn 21439 df-cnp 21440 df-lm 21441 df-haus 21527 df-tx 21774 df-hmeo 21967 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-xms 22533 df-ms 22534 df-tms 22535 df-cfil 23461 df-cau 23462 df-cmet 23463 df-grpo 27920 df-gid 27921 df-ginv 27922 df-gdiv 27923 df-ablo 27972 df-vc 27986 df-nv 28019 df-va 28022 df-ba 28023 df-sm 28024 df-0v 28025 df-vs 28026 df-nmcv 28027 df-ims 28028 df-dip 28128 df-ssp 28149 df-ph 28240 df-cbn 28291 df-hnorm 28397 df-hba 28398 df-hvsub 28400 df-hlim 28401 df-hcau 28402 df-sh 28636 df-ch 28650 df-oc 28681 df-ch0 28682 df-shs 28739 |
This theorem is referenced by: (None) |
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