Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihopelvalcqat | Structured version Visualization version GIF version |
Description: Ordered pair member of the partial isomorphism H for atom argument not under 𝑊. TODO: remove .t hypothesis. (Contributed by NM, 30-Mar-2014.) |
Ref | Expression |
---|---|
dihelval2.l | ⊢ ≤ = (le‘𝐾) |
dihelval2.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dihelval2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihelval2.p | ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
dihelval2.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dihelval2.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dihelval2.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihelval2.g | ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄) |
dihelval2.f | ⊢ 𝐹 ∈ V |
dihelval2.s | ⊢ 𝑆 ∈ V |
Ref | Expression |
---|---|
dihopelvalcqat | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑄) ↔ (𝐹 = (𝑆‘𝐺) ∧ 𝑆 ∈ 𝐸))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihelval2.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | dihelval2.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | dihelval2.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | eqid 2738 | . . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
5 | dihelval2.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dihvalcqat 38865 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = (((DIsoC‘𝐾)‘𝑊)‘𝑄)) |
7 | 6 | eleq2d 2818 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑄) ↔ 〈𝐹, 𝑆〉 ∈ (((DIsoC‘𝐾)‘𝑊)‘𝑄))) |
8 | dihelval2.p | . . 3 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
9 | dihelval2.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
10 | dihelval2.e | . . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
11 | dihelval2.g | . . 3 ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄) | |
12 | dihelval2.f | . . 3 ⊢ 𝐹 ∈ V | |
13 | dihelval2.s | . . 3 ⊢ 𝑆 ∈ V | |
14 | 1, 2, 3, 8, 9, 10, 4, 11, 12, 13 | dicopelval2 38807 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (((DIsoC‘𝐾)‘𝑊)‘𝑄) ↔ (𝐹 = (𝑆‘𝐺) ∧ 𝑆 ∈ 𝐸))) |
15 | 7, 14 | bitrd 282 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑄) ↔ (𝐹 = (𝑆‘𝐺) ∧ 𝑆 ∈ 𝐸))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2113 Vcvv 3397 〈cop 4519 class class class wbr 5027 ‘cfv 6333 ℩crio 7120 lecple 16668 occoc 16669 Atomscatm 36889 HLchlt 36976 LHypclh 37610 LTrncltrn 37727 TEndoctendo 38378 DIsoCcdic 38798 DIsoHcdih 38854 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-cnex 10664 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 ax-pre-mulgt0 10685 ax-riotaBAD 36579 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-iin 4881 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-riota 7121 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-tpos 7914 df-undef 7961 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-er 8313 df-map 8432 df-en 8549 df-dom 8550 df-sdom 8551 df-fin 8552 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-sub 10943 df-neg 10944 df-nn 11710 df-2 11772 df-3 11773 df-4 11774 df-5 11775 df-6 11776 df-n0 11970 df-z 12056 df-uz 12318 df-fz 12975 df-struct 16581 df-ndx 16582 df-slot 16583 df-base 16585 df-sets 16586 df-ress 16587 df-plusg 16674 df-mulr 16675 df-sca 16677 df-vsca 16678 df-0g 16811 df-proset 17647 df-poset 17665 df-plt 17677 df-lub 17693 df-glb 17694 df-join 17695 df-meet 17696 df-p0 17758 df-p1 17759 df-lat 17765 df-clat 17827 df-mgm 17961 df-sgrp 18010 df-mnd 18021 df-submnd 18066 df-grp 18215 df-minusg 18216 df-sbg 18217 df-subg 18387 df-cntz 18558 df-lsm 18872 df-cmn 19019 df-abl 19020 df-mgp 19352 df-ur 19364 df-ring 19411 df-oppr 19488 df-dvdsr 19506 df-unit 19507 df-invr 19537 df-dvr 19548 df-drng 19616 df-lmod 19748 df-lss 19816 df-lsp 19856 df-lvec 19987 df-oposet 36802 df-ol 36804 df-oml 36805 df-covers 36892 df-ats 36893 df-atl 36924 df-cvlat 36948 df-hlat 36977 df-llines 37124 df-lplanes 37125 df-lvols 37126 df-lines 37127 df-psubsp 37129 df-pmap 37130 df-padd 37422 df-lhyp 37614 df-laut 37615 df-ldil 37730 df-ltrn 37731 df-trl 37785 df-tendo 38381 df-edring 38383 df-disoa 38655 df-dvech 38705 df-dib 38765 df-dic 38799 df-dih 38855 |
This theorem is referenced by: dihmeetlem4preN 38932 dihmeetlem13N 38945 dihjatcclem4 39047 |
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