Nearby theorems
|
|
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
| Mirrors > Home > MPE Home > Th. List > Mathboxes > dihmeetcN | Structured version Visualization version GIF version | ||
| Description: Isomorphism H of a lattice meet when the meet is not under the fiducial hyperplane π. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| dihmeetc.b | β’ π΅ = (BaseβπΎ) |
| dihmeetc.l | β’ β€ = (leβπΎ) |
| dihmeetc.m | β’ β§ = (meetβπΎ) |
| dihmeetc.h | β’ π» = (LHypβπΎ) |
| dihmeetc.i | β’ πΌ = ((DIsoHβπΎ)βπ) |
| Ref | Expression |
|---|---|
| dihmeetcN | β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β (πΌβ(π β§ π)) = ((πΌβπ) β© (πΌβπ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2731 | . . . 4 β’ (glbβπΎ) = (glbβπΎ) | |
| 2 | dihmeetc.m | . . . 4 β’ β§ = (meetβπΎ) | |
| 3 | simp1l 1196 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β πΎ β HL) | |
| 4 | simp2l 1198 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β π β π΅) | |
| 5 | simp2r 1199 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β π β π΅) | |
| 6 | 1, 2, 3, 4, 5 | meetval 18349 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β (π β§ π) = ((glbβπΎ)β{π, π})) |
| 7 | 6 | fveq2d 6896 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β (πΌβ(π β§ π)) = (πΌβ((glbβπΎ)β{π, π}))) |
| 8 | simp1 1135 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β (πΎ β HL β§ π β π»)) | |
| 9 | prssi 4825 | . . . 4 β’ ((π β π΅ β§ π β π΅) β {π, π} β π΅) | |
| 10 | 9 | 3ad2ant2 1133 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β {π, π} β π΅) |
| 11 | prnzg 4783 | . . . 4 β’ (π β π΅ β {π, π} β β ) | |
| 12 | 4, 11 | syl 17 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β {π, π} β β ) |
| 13 | simp3 1137 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β Β¬ (π β§ π) β€ π) | |
| 14 | 6 | breq1d 5159 | . . . 4 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β ((π β§ π) β€ π β ((glbβπΎ)β{π, π}) β€ π)) |
| 15 | 13, 14 | mtbid 323 | . . 3 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β Β¬ ((glbβπΎ)β{π, π}) β€ π) |
| 16 | dihmeetc.b | . . . 4 β’ π΅ = (BaseβπΎ) | |
| 17 | dihmeetc.h | . . . 4 β’ π» = (LHypβπΎ) | |
| 18 | dihmeetc.i | . . . 4 β’ πΌ = ((DIsoHβπΎ)βπ) | |
| 19 | dihmeetc.l | . . . 4 β’ β€ = (leβπΎ) | |
| 20 | 16, 1, 17, 18, 19 | dihglbcN 40476 | . . 3 β’ (((πΎ β HL β§ π β π») β§ ({π, π} β π΅ β§ {π, π} β β ) β§ Β¬ ((glbβπΎ)β{π, π}) β€ π) β (πΌβ((glbβπΎ)β{π, π})) = β© π₯ β {π, π} (πΌβπ₯)) |
| 21 | 8, 10, 12, 15, 20 | syl121anc 1374 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β (πΌβ((glbβπΎ)β{π, π})) = β© π₯ β {π, π} (πΌβπ₯)) |
| 22 | fveq2 6892 | . . . 4 β’ (π₯ = π β (πΌβπ₯) = (πΌβπ)) | |
| 23 | fveq2 6892 | . . . 4 β’ (π₯ = π β (πΌβπ₯) = (πΌβπ)) | |
| 24 | 22, 23 | iinxprg 5093 | . . 3 β’ ((π β π΅ β§ π β π΅) β β© π₯ β {π, π} (πΌβπ₯) = ((πΌβπ) β© (πΌβπ))) |
| 25 | 24 | 3ad2ant2 1133 | . 2 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β β© π₯ β {π, π} (πΌβπ₯) = ((πΌβπ) β© (πΌβπ))) |
| 26 | 7, 21, 25 | 3eqtrd 2775 | 1 β’ (((πΎ β HL β§ π β π») β§ (π β π΅ β§ π β π΅) β§ Β¬ (π β§ π) β€ π) β (πΌβ(π β§ π)) = ((πΌβπ) β© (πΌβπ))) |
| Colors of variables: wff setvar class |
| Syntax hints: Β¬ wn 3 β wi 4 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 β wne 2939 β© cin 3948 β wss 3949 β c0 4323 {cpr 4631 β© ciin 4999 class class class wbr 5149 βcfv 6544 (class class class)co 7412 Basecbs 17149 lecple 17209 glbcglb 18268 meetcmee 18270 HLchlt 38524 LHypclh 39159 DIsoHcdih 40403 |
| 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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-riotaBAD 38127 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-tpos 8214 df-undef 8261 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-n0 12478 df-z 12564 df-uz 12828 df-fz 13490 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-sca 17218 df-vsca 17219 df-0g 17392 df-proset 18253 df-poset 18271 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18390 df-clat 18457 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-cntz 19223 df-lsm 19546 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20503 df-lmod 20617 df-lss 20688 df-lsp 20728 df-lvec 20859 df-oposet 38350 df-ol 38352 df-oml 38353 df-covers 38440 df-ats 38441 df-atl 38472 df-cvlat 38496 df-hlat 38525 df-llines 38673 df-lplanes 38674 df-lvols 38675 df-lines 38676 df-psubsp 38678 df-pmap 38679 df-padd 38971 df-lhyp 39163 df-laut 39164 df-ldil 39279 df-ltrn 39280 df-trl 39334 df-tendo 39930 df-edring 39932 df-disoa 40204 df-dvech 40254 df-dib 40314 df-dic 40348 df-dih 40404 |
| This theorem is referenced by: dihmeetlem10N 40491 dihmeetALTN 40502 |
| Copyright terms: Public domain | W3C validator |