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Mirrors > Home > MPE Home > Th. List > Mathboxes > dib2dim | Structured version Visualization version GIF version |
Description: Extend dia2dim 39472 to partial isomorphism B. (Contributed by NM, 22-Sep-2014.) |
Ref | Expression |
---|---|
dib2dim.l | ⊢ ≤ = (le‘𝐾) |
dib2dim.j | ⊢ ∨ = (join‘𝐾) |
dib2dim.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dib2dim.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dib2dim.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dib2dim.s | ⊢ ⊕ = (LSSum‘𝑈) |
dib2dim.i | ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) |
dib2dim.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dib2dim.p | ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) |
dib2dim.q | ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) |
Ref | Expression |
---|---|
dib2dim | ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dib2dim.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dib2dim.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
3 | dib2dim.i | . . . 4 ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) | |
4 | 2, 3 | dibvalrel 39558 | . . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → Rel (𝐼‘(𝑃 ∨ 𝑄))) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝜑 → Rel (𝐼‘(𝑃 ∨ 𝑄))) |
6 | dib2dim.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
7 | dib2dim.j | . . . . . 6 ⊢ ∨ = (join‘𝐾) | |
8 | dib2dim.a | . . . . . 6 ⊢ 𝐴 = (Atoms‘𝐾) | |
9 | eqid 2736 | . . . . . 6 ⊢ ((DVecA‘𝐾)‘𝑊) = ((DVecA‘𝐾)‘𝑊) | |
10 | eqid 2736 | . . . . . 6 ⊢ (LSSum‘((DVecA‘𝐾)‘𝑊)) = (LSSum‘((DVecA‘𝐾)‘𝑊)) | |
11 | eqid 2736 | . . . . . 6 ⊢ ((DIsoA‘𝐾)‘𝑊) = ((DIsoA‘𝐾)‘𝑊) | |
12 | dib2dim.p | . . . . . 6 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) | |
13 | dib2dim.q | . . . . . 6 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) | |
14 | 6, 7, 8, 2, 9, 10, 11, 1, 12, 13 | dia2dim 39472 | . . . . 5 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ⊆ ((((DIsoA‘𝐾)‘𝑊)‘𝑃)(LSSum‘((DVecA‘𝐾)‘𝑊))(((DIsoA‘𝐾)‘𝑊)‘𝑄))) |
15 | 14 | sseld 3941 | . . . 4 ⊢ (𝜑 → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) → 𝑓 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑃)(LSSum‘((DVecA‘𝐾)‘𝑊))(((DIsoA‘𝐾)‘𝑊)‘𝑄)))) |
16 | 15 | anim1d 611 | . . 3 ⊢ (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 495 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
18 | 12 | simpld 495 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
19 | 13 | simpld 495 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
20 | eqid 2736 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | 20, 7, 8 | hlatjcl 37761 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
22 | 17, 18, 19, 21 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
23 | 12 | simprd 496 | . . . . 5 ⊢ (𝜑 → 𝑃 ≤ 𝑊) |
24 | 13 | simprd 496 | . . . . 5 ⊢ (𝜑 → 𝑄 ≤ 𝑊) |
25 | 17 | hllatd 37758 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat) |
26 | 20, 8 | atbase 37683 | . . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
27 | 18, 26 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
28 | 20, 8 | atbase 37683 | . . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
29 | 19, 28 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾)) |
30 | 1 | simprd 496 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
31 | 20, 2 | lhpbase 38393 | . . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
32 | 30, 31 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
33 | 20, 6, 7 | latjle12 18293 | . . . . . 6 ⊢ ((𝐾 ∈ Lat ∧ (𝑃 ∈ (Base‘𝐾) ∧ 𝑄 ∈ (Base‘𝐾) ∧ 𝑊 ∈ (Base‘𝐾))) → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊)) |
34 | 25, 27, 29, 32, 33 | syl13anc 1372 | . . . . 5 ⊢ (𝜑 → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊)) |
35 | 23, 24, 34 | mpbi2and 710 | . . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ 𝑊) |
36 | eqid 2736 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
37 | eqid 2736 | . . . . 5 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
38 | 20, 6, 2, 36, 37, 11, 3 | dibopelval2 39540 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∨ 𝑄) ∈ (Base‘𝐾) ∧ (𝑃 ∨ 𝑄) ≤ 𝑊)) → (〈𝑓, 𝑠〉 ∈ (𝐼‘(𝑃 ∨ 𝑄)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ∧ 𝑠 = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))) |
39 | 1, 22, 35, 38 | syl12anc 835 | . . 3 ⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ (𝐼‘(𝑃 ∨ 𝑄)) ↔ (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ∧ 𝑠 = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))) |
40 | dib2dim.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
41 | dib2dim.s | . . . 4 ⊢ ⊕ = (LSSum‘𝑈) | |
42 | 27, 23 | jca 512 | . . . 4 ⊢ (𝜑 → (𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≤ 𝑊)) |
43 | 29, 24 | jca 512 | . . . 4 ⊢ (𝜑 → (𝑄 ∈ (Base‘𝐾) ∧ 𝑄 ≤ 𝑊)) |
44 | 20, 6, 2, 36, 37, 9, 40, 10, 41, 11, 3, 1, 42, 43 | diblsmopel 39566 | . . 3 ⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ↔ (𝑓 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑃)(LSSum‘((DVecA‘𝐾)‘𝑊))(((DIsoA‘𝐾)‘𝑊)‘𝑄)) ∧ 𝑠 = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾)))))) |
45 | 16, 39, 44 | 3imtr4d 293 | . 2 ⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ (𝐼‘(𝑃 ∨ 𝑄)) → 〈𝑓, 𝑠〉 ∈ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))) |
46 | 5, 45 | relssdv 5742 | 1 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 〈cop 4590 class class class wbr 5103 ↦ cmpt 5186 I cid 5528 ↾ cres 5633 Rel wrel 5636 ‘cfv 6493 (class class class)co 7351 Basecbs 17037 lecple 17094 joincjn 18154 Latclat 18274 LSSumclsm 19369 Atomscatm 37657 HLchlt 37744 LHypclh 38379 LTrncltrn 38496 DVecAcdveca 39397 DIsoAcdia 39423 DVecHcdvh 39473 DIsoBcdib 39533 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-riotaBAD 37347 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-undef 8196 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-struct 16973 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-mulr 17101 df-sca 17103 df-vsca 17104 df-0g 17277 df-proset 18138 df-poset 18156 df-plt 18173 df-lub 18189 df-glb 18190 df-join 18191 df-meet 18192 df-p0 18268 df-p1 18269 df-lat 18275 df-clat 18342 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-submnd 18556 df-grp 18705 df-minusg 18706 df-sbg 18707 df-subg 18878 df-cntz 19050 df-lsm 19371 df-cmn 19517 df-abl 19518 df-mgp 19850 df-ur 19867 df-ring 19914 df-oppr 19996 df-dvdsr 20017 df-unit 20018 df-invr 20048 df-dvr 20059 df-drng 20134 df-lmod 20271 df-lss 20340 df-lsp 20380 df-lvec 20511 df-oposet 37570 df-ol 37572 df-oml 37573 df-covers 37660 df-ats 37661 df-atl 37692 df-cvlat 37716 df-hlat 37745 df-llines 37893 df-lplanes 37894 df-lvols 37895 df-lines 37896 df-psubsp 37898 df-pmap 37899 df-padd 38191 df-lhyp 38383 df-laut 38384 df-ldil 38499 df-ltrn 38500 df-trl 38554 df-tgrp 39138 df-tendo 39150 df-edring 39152 df-dveca 39398 df-disoa 39424 df-dvech 39474 df-dib 39534 |
This theorem is referenced by: dih2dimb 39639 |
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