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Mirrors > Home > MPE Home > Th. List > Mathboxes > dib2dim | Structured version Visualization version GIF version |
Description: Extend dia2dim 38246 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 38332 | . . 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 2820 | . . . . . 6 ⊢ ((DVecA‘𝐾)‘𝑊) = ((DVecA‘𝐾)‘𝑊) | |
10 | eqid 2820 | . . . . . 6 ⊢ (LSSum‘((DVecA‘𝐾)‘𝑊)) = (LSSum‘((DVecA‘𝐾)‘𝑊)) | |
11 | eqid 2820 | . . . . . 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 38246 | . . . . 5 ⊢ (𝜑 → (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) ⊆ ((((DIsoA‘𝐾)‘𝑊)‘𝑃)(LSSum‘((DVecA‘𝐾)‘𝑊))(((DIsoA‘𝐾)‘𝑊)‘𝑄))) |
15 | 14 | sseld 3959 | . . . 4 ⊢ (𝜑 → (𝑓 ∈ (((DIsoA‘𝐾)‘𝑊)‘(𝑃 ∨ 𝑄)) → 𝑓 ∈ ((((DIsoA‘𝐾)‘𝑊)‘𝑃)(LSSum‘((DVecA‘𝐾)‘𝑊))(((DIsoA‘𝐾)‘𝑊)‘𝑄)))) |
16 | 15 | anim1d 612 | . . 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 497 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
18 | 12 | simpld 497 | . . . . 5 ⊢ (𝜑 → 𝑃 ∈ 𝐴) |
19 | 13 | simpld 497 | . . . . 5 ⊢ (𝜑 → 𝑄 ∈ 𝐴) |
20 | eqid 2820 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
21 | 20, 7, 8 | hlatjcl 36536 | . . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
22 | 17, 18, 19, 21 | syl3anc 1366 | . . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ∈ (Base‘𝐾)) |
23 | 12 | simprd 498 | . . . . 5 ⊢ (𝜑 → 𝑃 ≤ 𝑊) |
24 | 13 | simprd 498 | . . . . 5 ⊢ (𝜑 → 𝑄 ≤ 𝑊) |
25 | 17 | hllatd 36533 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ Lat) |
26 | 20, 8 | atbase 36458 | . . . . . . 7 ⊢ (𝑃 ∈ 𝐴 → 𝑃 ∈ (Base‘𝐾)) |
27 | 18, 26 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ (Base‘𝐾)) |
28 | 20, 8 | atbase 36458 | . . . . . . 7 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝐾)) |
29 | 19, 28 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑄 ∈ (Base‘𝐾)) |
30 | 1 | simprd 498 | . . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ 𝐻) |
31 | 20, 2 | lhpbase 37167 | . . . . . . 7 ⊢ (𝑊 ∈ 𝐻 → 𝑊 ∈ (Base‘𝐾)) |
32 | 30, 31 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐾)) |
33 | 20, 6, 7 | latjle12 17667 | . . . . . 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 1367 | . . . . 5 ⊢ (𝜑 → ((𝑃 ≤ 𝑊 ∧ 𝑄 ≤ 𝑊) ↔ (𝑃 ∨ 𝑄) ≤ 𝑊)) |
35 | 23, 24, 34 | mpbi2and 710 | . . . 4 ⊢ (𝜑 → (𝑃 ∨ 𝑄) ≤ 𝑊) |
36 | eqid 2820 | . . . . 5 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊) | |
37 | eqid 2820 | . . . . 5 ⊢ (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) = (𝑓 ∈ ((LTrn‘𝐾)‘𝑊) ↦ ( I ↾ (Base‘𝐾))) | |
38 | 20, 6, 2, 36, 37, 11, 3 | dibopelval2 38314 | . . . 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 834 | . . 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 514 | . . . 4 ⊢ (𝜑 → (𝑃 ∈ (Base‘𝐾) ∧ 𝑃 ≤ 𝑊)) |
43 | 29, 24 | jca 514 | . . . 4 ⊢ (𝜑 → (𝑄 ∈ (Base‘𝐾) ∧ 𝑄 ≤ 𝑊)) |
44 | 20, 6, 2, 36, 37, 9, 40, 10, 41, 11, 3, 1, 42, 43 | diblsmopel 38340 | . . 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 296 | . 2 ⊢ (𝜑 → (〈𝑓, 𝑠〉 ∈ (𝐼‘(𝑃 ∨ 𝑄)) → 〈𝑓, 𝑠〉 ∈ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))) |
46 | 5, 45 | relssdv 5654 | 1 ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ⊆ wss 3929 〈cop 4566 class class class wbr 5059 ↦ cmpt 5139 I cid 5452 ↾ cres 5550 Rel wrel 5553 ‘cfv 6348 (class class class)co 7149 Basecbs 16478 lecple 16567 joincjn 17549 Latclat 17650 LSSumclsm 18754 Atomscatm 36432 HLchlt 36519 LHypclh 37153 LTrncltrn 37270 DVecAcdveca 38171 DIsoAcdia 38197 DVecHcdvh 38247 DIsoBcdib 38307 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-riotaBAD 36122 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 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 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-1st 7682 df-2nd 7683 df-tpos 7885 df-undef 7932 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-1o 8095 df-oadd 8099 df-er 8282 df-map 8401 df-en 8503 df-dom 8504 df-sdom 8505 df-fin 8506 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 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 12890 df-struct 16480 df-ndx 16481 df-slot 16482 df-base 16484 df-sets 16485 df-ress 16486 df-plusg 16573 df-mulr 16574 df-sca 16576 df-vsca 16577 df-0g 16710 df-proset 17533 df-poset 17551 df-plt 17563 df-lub 17579 df-glb 17580 df-join 17581 df-meet 17582 df-p0 17644 df-p1 17645 df-lat 17651 df-clat 17713 df-mgm 17847 df-sgrp 17896 df-mnd 17907 df-submnd 17952 df-grp 18101 df-minusg 18102 df-sbg 18103 df-subg 18271 df-cntz 18442 df-lsm 18756 df-cmn 18903 df-abl 18904 df-mgp 19235 df-ur 19247 df-ring 19294 df-oppr 19368 df-dvdsr 19386 df-unit 19387 df-invr 19417 df-dvr 19428 df-drng 19499 df-lmod 19631 df-lss 19699 df-lsp 19739 df-lvec 19870 df-oposet 36345 df-ol 36347 df-oml 36348 df-covers 36435 df-ats 36436 df-atl 36467 df-cvlat 36491 df-hlat 36520 df-llines 36667 df-lplanes 36668 df-lvols 36669 df-lines 36670 df-psubsp 36672 df-pmap 36673 df-padd 36965 df-lhyp 37157 df-laut 37158 df-ldil 37273 df-ltrn 37274 df-trl 37328 df-tgrp 37912 df-tendo 37924 df-edring 37926 df-dveca 38172 df-disoa 38198 df-dvech 38248 df-dib 38308 |
This theorem is referenced by: dih2dimb 38413 |
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