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Mirrors > Home > MPE Home > Th. List > Mathboxes > dih2dimbALTN | Structured version Visualization version GIF version |
Description: Extend dia2dim 38207 to isomorphism H. (This version combines dib2dim 38373 and dih2dimb 38374 for shorter overall proof, but may be less easy to understand. TODO: decide which to use.) (Contributed by NM, 22-Sep-2014.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dih2dimb.l | ⊢ ≤ = (le‘𝐾) |
dih2dimb.j | ⊢ ∨ = (join‘𝐾) |
dih2dimb.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dih2dimb.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dih2dimb.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dih2dimb.s | ⊢ ⊕ = (LSSum‘𝑈) |
dih2dimb.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dih2dimb.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dih2dimb.p | ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) |
dih2dimb.q | ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) |
Ref | Expression |
---|---|
dih2dimbALTN | ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dih2dimb.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dih2dimb.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | eqid 2821 | . . . . 5 ⊢ ((DIsoB‘𝐾)‘𝑊) = ((DIsoB‘𝐾)‘𝑊) | |
4 | 2, 3 | dibvalrel 38293 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (((DIsoB‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄))) |
5 | 1, 4 | syl 17 | . . 3 ⊢ (𝜑 → Rel (((DIsoB‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄))) |
6 | dih2dimb.l | . . . . . . 7 ⊢ ≤ = (le‘𝐾) | |
7 | dih2dimb.j | . . . . . . 7 ⊢ ∨ = (join‘𝐾) | |
8 | dih2dimb.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | eqid 2821 | . . . . . . 7 ⊢ ((DVecA‘𝐾)‘𝑊) = ((DVecA‘𝐾)‘𝑊) | |
10 | eqid 2821 | . . . . . . 7 ⊢ (LSSum‘((DVecA‘𝐾)‘𝑊)) = (LSSum‘((DVecA‘𝐾)‘𝑊)) | |
11 | eqid 2821 | . . . . . . 7 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
12 | dih2dimb.p | . . . . . . 7 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) | |
13 | dih2dimb.q | . . . . . . 7 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) | |
14 | 6, 7, 8, 2, 9, 10, 11, 1, 12, 13 | dia2dim 38207 | . . . . . 6 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ⊆ ((((DIsoA‘𝐾)‘𝑊)‘𝑃)(LSSum‘((DVecA‘𝐾)‘𝑊))(((DIsoA‘𝐾)‘𝑊)‘𝑄))) |
15 | 14 | sseld 3966 | . . . . 5 ⊢ (𝜑 → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) → 𝑓 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑃)(LSSum‘((DVecA‘𝐾)‘𝑊))(((DIsoA‘𝐾)‘𝑊)‘𝑄)))) |
16 | 15 | anim1d 612 | . . . 4 ⊢ (𝜑 → ((𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ∧ 𝑠 = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))) → (𝑓 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑃)(LSSum‘((DVecA‘𝐾)‘𝑊))(((DIsoA‘𝐾)‘𝑊)‘𝑄)) ∧ 𝑠 = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))) |
17 | 1 | simpld 497 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ HL) |
18 | 12 | simpld 497 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
19 | 13 | simpld 497 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
20 | eqid 2821 | . . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | 20, 7, 8 | hlatjcl 36497 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
22 | 17, 18, 19, 21 | syl3anc 1367 | . . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
23 | 12 | simprd 498 | . . . . . 6 ⊢ (𝜑 → 𝑃 ≤ 𝑊) |
24 | 13 | simprd 498 | . . . . . 6 ⊢ (𝜑 → 𝑄 ≤ 𝑊) |
25 | hllat 36493 | . . . . . . . 8 ⊢ (𝐾 ∈ HL → 𝐾 ∈ Lat) | |
26 | 17, 25 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ Lat) |
27 | 20, 8 | atbase 36419 | . . . . . . . 8 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
28 | 18, 27 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
29 | 20, 8 | atbase 36419 | . . . . . . . 8 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
30 | 19, 29 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾)) |
31 | 1 | simprd 498 | . . . . . . . 8 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
32 | 20, 2 | lhpbase 37128 | . . . . . . . 8 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
33 | 31, 32 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
34 | 20, 6, 7 | latjle12 17666 | . . . . . . 7 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊)) |
35 | 26, 28, 30, 33, 34 | syl13anc 1368 | . . . . . 6 ⊢ (𝜑 → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊)) |
36 | 23, 24, 35 | mpbi2and 710 | . . . . 5 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ 𝑊) |
37 | eqid 2821 | . . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
38 | eqid 2821 | . . . . . 6 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
39 | 20, 6, 2, 37, 38, 11, 3 | dibopelval2 38275 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ≤ 𝑊)) → (〈𝑓, 𝑠〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ∧ 𝑠 = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))) |
40 | 1, 22, 36, 39 | syl12anc 834 | . . . 4 ⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ∧ 𝑠 = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))) |
41 | dih2dimb.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
42 | dih2dimb.s | . . . . 5 ⊢ ⊕ = (LSSum‘𝑈) | |
43 | 28, 23 | jca 514 | . . . . 5 ⊢ (𝜑 → (𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≤ 𝑊)) |
44 | 30, 24 | jca 514 | . . . . 5 ⊢ (𝜑 → (𝑄 ∈ (Base‘𝐾) ∧ 𝑄 ≤ 𝑊)) |
45 | 20, 6, 2, 37, 38, 9, 41, 10, 42, 11, 3, 1, 43, 44 | diblsmopel 38301 | . . . 4 ⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ ((((DIsoB‘𝐾)‘𝑊)‘𝑃) ⊕ (((DIsoB‘𝐾)‘𝑊)‘𝑄)) ↔ (𝑓 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑃)(LSSum‘((DVecA‘𝐾)‘𝑊))(((DIsoA‘𝐾)‘𝑊)‘𝑄)) ∧ 𝑠 = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))) |
46 | 16, 40, 45 | 3imtr4d 296 | . . 3 ⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ (((DIsoB‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) → 〈𝑓, 𝑠〉 ∈ ((((DIsoB‘𝐾)‘𝑊)‘𝑃) ⊕ (((DIsoB‘𝐾)‘𝑊)‘𝑄)))) |
47 | 5, 46 | relssdv 5656 | . 2 ⊢ (𝜑 → (((DIsoB‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ⊆ ((((DIsoB‘𝐾)‘𝑊)‘𝑃) ⊕ (((DIsoB‘𝐾)‘𝑊)‘𝑄))) |
48 | dih2dimb.i | . . . 4 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
49 | 20, 6, 2, 48, 3 | dihvalb 38367 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ≤ 𝑊)) → (𝐼‘(𝑃 ∨ 𝑄)) = (((DIsoB‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄))) |
50 | 1, 22, 36, 49 | syl12anc 834 | . 2 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((DIsoB‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄))) |
51 | 20, 6, 2, 48, 3 | dihvalb 38367 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≤ 𝑊)) → (𝐼‘𝑃) = (((DIsoB‘𝐾)‘𝑊)‘𝑃)) |
52 | 1, 28, 23, 51 | syl12anc 834 | . . 3 ⊢ (𝜑 → (𝐼‘𝑃) = (((DIsoB‘𝐾)‘𝑊)‘𝑃)) |
53 | 20, 6, 2, 48, 3 | dihvalb 38367 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ (Base‘𝐾) ∧ 𝑄 ≤ 𝑊)) → (𝐼‘𝑄) = (((DIsoB‘𝐾)‘𝑊)‘𝑄)) |
54 | 1, 30, 24, 53 | syl12anc 834 | . . 3 ⊢ (𝜑 → (𝐼‘𝑄) = (((DIsoB‘𝐾)‘𝑊)‘𝑄)) |
55 | 52, 54 | oveq12d 7168 | . 2 ⊢ (𝜑 → ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) = ((((DIsoB‘𝐾)‘𝑊)‘𝑃) ⊕ (((DIsoB‘𝐾)‘𝑊)‘𝑄))) |
56 | 47, 50, 55 | 3sstr4d 4014 | 1 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 〈cop 4567 class class class wbr 5059 ↦ cmpt 5139 I cid 5454 ↾ cres 5552 Rel wrel 5555 ‘cfv 6350 (class class class)co 7150 Basecbs 16477 lecple 16566 joincjn 17548 Latclat 17649 LSSumclsm 18753 Atomscatm 36393 HLchlt 36480 LHypclh 37114 LTrncltrn 37231 DVecAcdveca 38132 DIsoAcdia 38158 DVecHcdvh 38208 DIsoBcdib 38268 DIsoHcdih 38358 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-riotaBAD 36083 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-tpos 7886 df-undef 7933 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-n0 11892 df-z 11976 df-uz 12238 df-fz 12887 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-sca 16575 df-vsca 16576 df-0g 16709 df-proset 17532 df-poset 17550 df-plt 17562 df-lub 17578 df-glb 17579 df-join 17580 df-meet 17581 df-p0 17643 df-p1 17644 df-lat 17650 df-clat 17712 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-subg 18270 df-cntz 18441 df-lsm 18755 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-lmod 19630 df-lss 19698 df-lsp 19738 df-lvec 19869 df-oposet 36306 df-ol 36308 df-oml 36309 df-covers 36396 df-ats 36397 df-atl 36428 df-cvlat 36452 df-hlat 36481 df-llines 36628 df-lplanes 36629 df-lvols 36630 df-lines 36631 df-psubsp 36633 df-pmap 36634 df-padd 36926 df-lhyp 37118 df-laut 37119 df-ldil 37234 df-ltrn 37235 df-trl 37289 df-tgrp 37873 df-tendo 37885 df-edring 37887 df-dveca 38133 df-disoa 38159 df-dvech 38209 df-dib 38269 df-dih 38359 |
This theorem is referenced by: (None) |
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