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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 𝑊. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dihmeetc.b | ⊢ 𝐵 = (Base‘𝐾) |
dihmeetc.l | ⊢ ≤ = (le‘𝐾) |
dihmeetc.m | ⊢ ∧ = (meet‘𝐾) |
dihmeetc.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihmeetc.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
Ref | Expression |
---|---|
dihmeetcN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2726 | . . . 4 ⊢ (glb‘𝐾) = (glb‘𝐾) | |
2 | dihmeetc.m | . . . 4 ⊢ ∧ = (meet‘𝐾) | |
3 | simp1l 1194 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → 𝐾 ∈ HL) | |
4 | simp2l 1196 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → 𝑋 ∈ 𝐵) | |
5 | simp2r 1197 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → 𝑌 ∈ 𝐵) | |
6 | 1, 2, 3, 4, 5 | meetval 18416 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → (𝑋 ∧ 𝑌) = ((glb‘𝐾)‘{𝑋, 𝑌})) |
7 | 6 | fveq2d 6905 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → (𝐼‘(𝑋 ∧ 𝑌)) = (𝐼‘((glb‘𝐾)‘{𝑋, 𝑌}))) |
8 | simp1 1133 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | prssi 4830 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → {𝑋, 𝑌} ⊆ 𝐵) | |
10 | 9 | 3ad2ant2 1131 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → {𝑋, 𝑌} ⊆ 𝐵) |
11 | prnzg 4787 | . . . 4 ⊢ (𝑋 ∈ 𝐵 → {𝑋, 𝑌} ≠ ∅) | |
12 | 4, 11 | syl 17 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → {𝑋, 𝑌} ≠ ∅) |
13 | simp3 1135 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) | |
14 | 6 | breq1d 5163 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → ((𝑋 ∧ 𝑌) ≤ 𝑊 ↔ ((glb‘𝐾)‘{𝑋, 𝑌}) ≤ 𝑊)) |
15 | 13, 14 | mtbid 323 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → ¬ ((glb‘𝐾)‘{𝑋, 𝑌}) ≤ 𝑊) |
16 | dihmeetc.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
17 | dihmeetc.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
18 | dihmeetc.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
19 | dihmeetc.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
20 | 16, 1, 17, 18, 19 | dihglbcN 41000 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ({𝑋, 𝑌} ⊆ 𝐵 ∧ {𝑋, 𝑌} ≠ ∅) ∧ ¬ ((glb‘𝐾)‘{𝑋, 𝑌}) ≤ 𝑊) → (𝐼‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥)) |
21 | 8, 10, 12, 15, 20 | syl121anc 1372 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → (𝐼‘((glb‘𝐾)‘{𝑋, 𝑌})) = ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥)) |
22 | fveq2 6901 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝐼‘𝑥) = (𝐼‘𝑋)) | |
23 | fveq2 6901 | . . . 4 ⊢ (𝑥 = 𝑌 → (𝐼‘𝑥) = (𝐼‘𝑌)) | |
24 | 22, 23 | iinxprg 5097 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) |
25 | 24 | 3ad2ant2 1131 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → ∩ 𝑥 ∈ {𝑋, 𝑌} (𝐼‘𝑥) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) |
26 | 7, 21, 25 | 3eqtrd 2770 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∩ cin 3946 ⊆ wss 3947 ∅c0 4325 {cpr 4635 ∩ ciin 5002 class class class wbr 5153 ‘cfv 6554 (class class class)co 7424 Basecbs 17213 lecple 17273 glbcglb 18335 meetcmee 18337 HLchlt 39048 LHypclh 39683 DIsoHcdih 40927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235 ax-riotaBAD 38651 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-tp 4638 df-op 4640 df-uni 4914 df-int 4955 df-iun 5003 df-iin 5004 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6312 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-om 7877 df-1st 8003 df-2nd 8004 df-tpos 8241 df-undef 8288 df-frecs 8296 df-wrecs 8327 df-recs 8401 df-rdg 8440 df-1o 8496 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-fin 8978 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12611 df-uz 12875 df-fz 13539 df-struct 17149 df-sets 17166 df-slot 17184 df-ndx 17196 df-base 17214 df-ress 17243 df-plusg 17279 df-mulr 17280 df-sca 17282 df-vsca 17283 df-0g 17456 df-proset 18320 df-poset 18338 df-plt 18355 df-lub 18371 df-glb 18372 df-join 18373 df-meet 18374 df-p0 18450 df-p1 18451 df-lat 18457 df-clat 18524 df-mgm 18633 df-sgrp 18712 df-mnd 18728 df-submnd 18774 df-grp 18931 df-minusg 18932 df-sbg 18933 df-subg 19117 df-cntz 19311 df-lsm 19634 df-cmn 19780 df-abl 19781 df-mgp 20118 df-rng 20136 df-ur 20165 df-ring 20218 df-oppr 20316 df-dvdsr 20339 df-unit 20340 df-invr 20370 df-dvr 20383 df-drng 20709 df-lmod 20838 df-lss 20909 df-lsp 20949 df-lvec 21081 df-oposet 38874 df-ol 38876 df-oml 38877 df-covers 38964 df-ats 38965 df-atl 38996 df-cvlat 39020 df-hlat 39049 df-llines 39197 df-lplanes 39198 df-lvols 39199 df-lines 39200 df-psubsp 39202 df-pmap 39203 df-padd 39495 df-lhyp 39687 df-laut 39688 df-ldil 39803 df-ltrn 39804 df-trl 39858 df-tendo 40454 df-edring 40456 df-disoa 40728 df-dvech 40778 df-dib 40838 df-dic 40872 df-dih 40928 |
This theorem is referenced by: dihmeetlem10N 41015 dihmeetALTN 41026 |
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