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 2821 | . . . 4 ⊢ ((DIsoC‘𝐾)‘𝑊) = ((DIsoC‘𝐾)‘𝑊) | |
5 | dihelval2.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | 1, 2, 3, 4, 5 | dihvalcqat 38390 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = (((DIsoC‘𝐾)‘𝑊)‘𝑄)) |
7 | 6 | eleq2d 2898 | . 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 38332 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (((DIsoC‘𝐾)‘𝑊)‘𝑄) ↔ (𝐹 = (𝑆‘𝐺) ∧ 𝑆 ∈ 𝐸))) |
15 | 7, 14 | bitrd 281 | 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 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 〈cop 4573 class class class wbr 5066 ‘cfv 6355 ℩crio 7113 lecple 16572 occoc 16573 Atomscatm 36414 HLchlt 36501 LHypclh 37135 LTrncltrn 37252 TEndoctendo 37903 DIsoCcdic 38323 DIsoHcdih 38379 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-0g 16715 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tendo 37906 df-edring 37908 df-disoa 38180 df-dvech 38230 df-dib 38290 df-dic 38324 df-dih 38380 |
This theorem is referenced by: dihmeetlem4preN 38457 dihmeetlem13N 38470 dihjatcclem4 38572 |
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