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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihmeet2 | Structured version Visualization version GIF version |
Description: Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.) |
Ref | Expression |
---|---|
dihmeet2.m | ⊢ ∧ = (meet‘𝐾) |
dihmeet2.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihmeet2.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihmeet2.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihmeet2.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
dihmeet2.y | ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) |
Ref | Expression |
---|---|
dihmeet2 | ⊢ (𝜑 → (◡𝐼‘(𝑋 ∩ 𝑌)) = ((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihmeet2.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dihmeet2.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
3 | dihmeet2.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | dihmeet2.i | . . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
5 | 3, 4 | dihcnvid2 40746 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝐼‘(◡𝐼‘𝑋)) = 𝑋) |
6 | 1, 2, 5 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝐼‘(◡𝐼‘𝑋)) = 𝑋) |
7 | dihmeet2.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) | |
8 | 3, 4 | dihcnvid2 40746 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → (𝐼‘(◡𝐼‘𝑌)) = 𝑌) |
9 | 1, 7, 8 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (𝐼‘(◡𝐼‘𝑌)) = 𝑌) |
10 | 6, 9 | ineq12d 4213 | . . 3 ⊢ (𝜑 → ((𝐼‘(◡𝐼‘𝑋)) ∩ (𝐼‘(◡𝐼‘𝑌))) = (𝑋 ∩ 𝑌)) |
11 | eqid 2728 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
12 | 11, 3, 4 | dihcnvcl 40744 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
13 | 1, 2, 12 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
14 | 11, 3, 4 | dihcnvcl 40744 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → (◡𝐼‘𝑌) ∈ (Base‘𝐾)) |
15 | 1, 7, 14 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (◡𝐼‘𝑌) ∈ (Base‘𝐾)) |
16 | dihmeet2.m | . . . . 5 ⊢ ∧ = (meet‘𝐾) | |
17 | 11, 16, 3, 4 | dihmeet 40816 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (◡𝐼‘𝑋) ∈ (Base‘𝐾) ∧ (◡𝐼‘𝑌) ∈ (Base‘𝐾)) → (𝐼‘((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌))) = ((𝐼‘(◡𝐼‘𝑋)) ∩ (𝐼‘(◡𝐼‘𝑌)))) |
18 | 1, 13, 15, 17 | syl3anc 1369 | . . 3 ⊢ (𝜑 → (𝐼‘((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌))) = ((𝐼‘(◡𝐼‘𝑋)) ∩ (𝐼‘(◡𝐼‘𝑌)))) |
19 | 3, 4 | dihmeetcl 40818 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ ran 𝐼 ∧ 𝑌 ∈ ran 𝐼)) → (𝑋 ∩ 𝑌) ∈ ran 𝐼) |
20 | 1, 2, 7, 19 | syl12anc 836 | . . . 4 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ∈ ran 𝐼) |
21 | 3, 4 | dihcnvid2 40746 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∩ 𝑌) ∈ ran 𝐼) → (𝐼‘(◡𝐼‘(𝑋 ∩ 𝑌))) = (𝑋 ∩ 𝑌)) |
22 | 1, 20, 21 | syl2anc 583 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘(𝑋 ∩ 𝑌))) = (𝑋 ∩ 𝑌)) |
23 | 10, 18, 22 | 3eqtr4rd 2779 | . 2 ⊢ (𝜑 → (𝐼‘(◡𝐼‘(𝑋 ∩ 𝑌))) = (𝐼‘((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌)))) |
24 | 11, 3, 4 | dihcnvcl 40744 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∩ 𝑌) ∈ ran 𝐼) → (◡𝐼‘(𝑋 ∩ 𝑌)) ∈ (Base‘𝐾)) |
25 | 1, 20, 24 | syl2anc 583 | . . 3 ⊢ (𝜑 → (◡𝐼‘(𝑋 ∩ 𝑌)) ∈ (Base‘𝐾)) |
26 | 1 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
27 | 26 | hllatd 38836 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat) |
28 | 11, 16 | latmcl 18432 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (◡𝐼‘𝑋) ∈ (Base‘𝐾) ∧ (◡𝐼‘𝑌) ∈ (Base‘𝐾)) → ((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌)) ∈ (Base‘𝐾)) |
29 | 27, 13, 15, 28 | syl3anc 1369 | . . 3 ⊢ (𝜑 → ((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌)) ∈ (Base‘𝐾)) |
30 | 11, 3, 4 | dih11 40738 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (◡𝐼‘(𝑋 ∩ 𝑌)) ∈ (Base‘𝐾) ∧ ((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌)) ∈ (Base‘𝐾)) → ((𝐼‘(◡𝐼‘(𝑋 ∩ 𝑌))) = (𝐼‘((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌))) ↔ (◡𝐼‘(𝑋 ∩ 𝑌)) = ((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌)))) |
31 | 1, 25, 29, 30 | syl3anc 1369 | . 2 ⊢ (𝜑 → ((𝐼‘(◡𝐼‘(𝑋 ∩ 𝑌))) = (𝐼‘((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌))) ↔ (◡𝐼‘(𝑋 ∩ 𝑌)) = ((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌)))) |
32 | 23, 31 | mpbid 231 | 1 ⊢ (𝜑 → (◡𝐼‘(𝑋 ∩ 𝑌)) = ((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∩ cin 3946 ◡ccnv 5677 ran crn 5679 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 meetcmee 18304 Latclat 18423 HLchlt 38822 LHypclh 39457 DIsoHcdih 40701 |
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 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-riotaBAD 38425 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-n0 12504 df-z 12590 df-uz 12854 df-fz 13518 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-sca 17249 df-vsca 17250 df-0g 17423 df-proset 18287 df-poset 18305 df-plt 18322 df-lub 18338 df-glb 18339 df-join 18340 df-meet 18341 df-p0 18417 df-p1 18418 df-lat 18424 df-clat 18491 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-submnd 18741 df-grp 18893 df-minusg 18894 df-sbg 18895 df-subg 19078 df-cntz 19268 df-lsm 19591 df-cmn 19737 df-abl 19738 df-mgp 20075 df-rng 20093 df-ur 20122 df-ring 20175 df-oppr 20273 df-dvdsr 20296 df-unit 20297 df-invr 20327 df-dvr 20340 df-drng 20626 df-lmod 20745 df-lss 20816 df-lsp 20856 df-lvec 20988 df-lsatoms 38448 df-oposet 38648 df-ol 38650 df-oml 38651 df-covers 38738 df-ats 38739 df-atl 38770 df-cvlat 38794 df-hlat 38823 df-llines 38971 df-lplanes 38972 df-lvols 38973 df-lines 38974 df-psubsp 38976 df-pmap 38977 df-padd 39269 df-lhyp 39461 df-laut 39462 df-ldil 39577 df-ltrn 39578 df-trl 39632 df-tendo 40228 df-edring 40230 df-disoa 40502 df-dvech 40552 df-dib 40612 df-dic 40646 df-dih 40702 |
This theorem is referenced by: dihoml4c 40849 |
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