Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > djhljjN | Structured version Visualization version GIF version |
Description: Lattice join in terms of DVecH vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
djhlj.b | ⊢ 𝐵 = (Base‘𝐾) |
djhlj.k | ⊢ ∨ = (join‘𝐾) |
djhlj.h | ⊢ 𝐻 = (LHyp‘𝐾) |
djhlj.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
djhlj.j | ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊) |
djhljj.w | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
djhljj.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
djhljj.y | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
djhljjN | ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | djhljj.w | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | djhljj.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | djhljj.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | djhlj.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
5 | djhlj.k | . . . . 5 ⊢ ∨ = (join‘𝐾) | |
6 | djhlj.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
7 | djhlj.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
8 | djhlj.j | . . . . 5 ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊) | |
9 | 4, 5, 6, 7, 8 | djhlj 38529 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌))) |
10 | 1, 2, 3, 9 | syl12anc 834 | . . 3 ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌))) |
11 | 4, 6, 7 | dihcl 38398 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ ran 𝐼) |
12 | 1, 2, 11 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝐼‘𝑋) ∈ ran 𝐼) |
13 | eqid 2819 | . . . . . . 7 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
14 | eqid 2819 | . . . . . . 7 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊)) | |
15 | 6, 13, 7, 14 | dihrnss 38406 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑋) ∈ ran 𝐼) → (𝐼‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
16 | 1, 12, 15 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝐼‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
17 | 4, 6, 7 | dihcl 38398 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵) → (𝐼‘𝑌) ∈ ran 𝐼) |
18 | 1, 3, 17 | syl2anc 586 | . . . . . 6 ⊢ (𝜑 → (𝐼‘𝑌) ∈ ran 𝐼) |
19 | 6, 13, 7, 14 | dihrnss 38406 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐼‘𝑌) ∈ ran 𝐼) → (𝐼‘𝑌) ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
20 | 1, 18, 19 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → (𝐼‘𝑌) ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
21 | 6, 7, 13, 14, 8 | djhcl 38528 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐼‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ (𝐼‘𝑌) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)))) → ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) ∈ ran 𝐼) |
22 | 1, 16, 20, 21 | syl12anc 834 | . . . 4 ⊢ (𝜑 → ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) ∈ ran 𝐼) |
23 | 6, 7 | dihcnvid2 38401 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) ∈ ran 𝐼) → (𝐼‘(◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌))) |
24 | 1, 22, 23 | syl2anc 586 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌))) |
25 | 10, 24 | eqtr4d 2857 | . 2 ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑌)) = (𝐼‘(◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))))) |
26 | 1 | simpld 497 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
27 | 26 | hllatd 36492 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat) |
28 | 4, 5 | latjcl 17653 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ∨ 𝑌) ∈ 𝐵) |
29 | 27, 2, 3, 28 | syl3anc 1366 | . . 3 ⊢ (𝜑 → (𝑋 ∨ 𝑌) ∈ 𝐵) |
30 | 4, 6, 7 | dihcnvcl 38399 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝐼‘𝑋)𝐽(𝐼‘𝑌)) ∈ ran 𝐼) → (◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))) ∈ 𝐵) |
31 | 1, 22, 30 | syl2anc 586 | . . 3 ⊢ (𝜑 → (◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))) ∈ 𝐵) |
32 | 4, 6, 7 | dih11 38393 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∨ 𝑌) ∈ 𝐵 ∧ (◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))) ∈ 𝐵) → ((𝐼‘(𝑋 ∨ 𝑌)) = (𝐼‘(◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) ↔ (𝑋 ∨ 𝑌) = (◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))))) |
33 | 1, 29, 31, 32 | syl3anc 1366 | . 2 ⊢ (𝜑 → ((𝐼‘(𝑋 ∨ 𝑌)) = (𝐼‘(◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) ↔ (𝑋 ∨ 𝑌) = (◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))))) |
34 | 25, 33 | mpbid 234 | 1 ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∈ wcel 2108 ⊆ wss 3934 ◡ccnv 5547 ran crn 5549 ‘cfv 6348 (class class class)co 7148 Basecbs 16475 joincjn 17546 Latclat 17647 HLchlt 36478 LHypclh 37112 DVecHcdvh 38206 DIsoHcdih 38356 joinHcdjh 38522 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-riotaBAD 36081 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-tpos 7884 df-undef 7931 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19232 df-ur 19244 df-ring 19291 df-oppr 19365 df-dvdsr 19383 df-unit 19384 df-invr 19414 df-dvr 19425 df-drng 19496 df-lmod 19628 df-lss 19696 df-lsp 19736 df-lvec 19867 df-lsatoms 36104 df-oposet 36304 df-ol 36306 df-oml 36307 df-covers 36394 df-ats 36395 df-atl 36426 df-cvlat 36450 df-hlat 36479 df-llines 36626 df-lplanes 36627 df-lvols 36628 df-lines 36629 df-psubsp 36631 df-pmap 36632 df-padd 36924 df-lhyp 37116 df-laut 37117 df-ldil 37232 df-ltrn 37233 df-trl 37287 df-tendo 37883 df-edring 37885 df-disoa 38157 df-dvech 38207 df-dib 38267 df-dic 38301 df-dih 38357 df-doch 38476 df-djh 38523 |
This theorem is referenced by: (None) |
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