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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihprrnlem1N | Structured version Visualization version GIF version |
Description: Lemma for dihprrn 40894, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
dihprrn.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihprrn.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihprrn.v | ⊢ 𝑉 = (Base‘𝑈) |
dihprrn.n | ⊢ 𝑁 = (LSpan‘𝑈) |
dihprrn.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dihprrn.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dihprrn.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
dihprrn.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
dihprrnlem1.l | ⊢ ≤ = (le‘𝐾) |
dihprrnlem1.o | ⊢ 0 = (0g‘𝑈) |
dihprrnlem1.nz | ⊢ (𝜑 → 𝑌 ≠ 0 ) |
dihprrnlem1.x | ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑋})) ≤ 𝑊) |
dihprrnlem1.y | ⊢ (𝜑 → ¬ (◡𝐼‘(𝑁‘{𝑌})) ≤ 𝑊) |
Ref | Expression |
---|---|
dihprrnlem1N | ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4628 | . . . 4 ⊢ {𝑋, 𝑌} = ({𝑋} ∪ {𝑌}) | |
2 | 1 | fveq2i 6895 | . . 3 ⊢ (𝑁‘{𝑋, 𝑌}) = (𝑁‘({𝑋} ∪ {𝑌})) |
3 | eqid 2728 | . . . . 5 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
4 | dihprrnlem1.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
5 | dihprrn.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | eqid 2728 | . . . . 5 ⊢ (join‘𝐾) = (join‘𝐾) | |
7 | eqid 2728 | . . . . 5 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾) | |
8 | dihprrn.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
9 | eqid 2728 | . . . . 5 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
10 | dihprrn.i | . . . . 5 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
11 | dihprrn.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | dihprrn.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
13 | dihprrn.v | . . . . . . . . 9 ⊢ 𝑉 = (Base‘𝑈) | |
14 | dihprrn.n | . . . . . . . . 9 ⊢ 𝑁 = (LSpan‘𝑈) | |
15 | 5, 8, 13, 14, 10 | dihlsprn 40799 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼) |
16 | 11, 12, 15 | syl2anc 583 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran 𝐼) |
17 | 3, 5, 10 | dihcnvcl 40739 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾)) |
18 | 11, 16, 17 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾)) |
19 | dihprrnlem1.x | . . . . . 6 ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑋})) ≤ 𝑊) | |
20 | 18, 19 | jca 511 | . . . . 5 ⊢ (𝜑 → ((◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾) ∧ (◡𝐼‘(𝑁‘{𝑋})) ≤ 𝑊)) |
21 | dihprrn.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
22 | dihprrnlem1.nz | . . . . . . 7 ⊢ (𝜑 → 𝑌 ≠ 0 ) | |
23 | dihprrnlem1.o | . . . . . . . 8 ⊢ 0 = (0g‘𝑈) | |
24 | 7, 5, 8, 13, 23, 14, 10 | dihlspsnat 40801 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ) → (◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾)) |
25 | 11, 21, 22, 24 | syl3anc 1369 | . . . . . 6 ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾)) |
26 | dihprrnlem1.y | . . . . . 6 ⊢ (𝜑 → ¬ (◡𝐼‘(𝑁‘{𝑌})) ≤ 𝑊) | |
27 | 25, 26 | jca 511 | . . . . 5 ⊢ (𝜑 → ((◡𝐼‘(𝑁‘{𝑌})) ∈ (Atoms‘𝐾) ∧ ¬ (◡𝐼‘(𝑁‘{𝑌})) ≤ 𝑊)) |
28 | 3, 4, 5, 6, 7, 8, 9, 10, 11, 20, 27 | dihjatc 40885 | . . . 4 ⊢ (𝜑 → (𝐼‘((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌})))) = ((𝐼‘(◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(◡𝐼‘(𝑁‘{𝑌}))))) |
29 | 5, 10 | dihcnvid2 40741 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran 𝐼) → (𝐼‘(◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋})) |
30 | 11, 16, 29 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝐼‘(◡𝐼‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋})) |
31 | 5, 8, 13, 14, 10 | dihlsprn 40799 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ ran 𝐼) |
32 | 11, 21, 31 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ ran 𝐼) |
33 | 5, 10 | dihcnvid2 40741 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (𝐼‘(◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌})) |
34 | 11, 32, 33 | syl2anc 583 | . . . . 5 ⊢ (𝜑 → (𝐼‘(◡𝐼‘(𝑁‘{𝑌}))) = (𝑁‘{𝑌})) |
35 | 30, 34 | oveq12d 7433 | . . . 4 ⊢ (𝜑 → ((𝐼‘(◡𝐼‘(𝑁‘{𝑋})))(LSSum‘𝑈)(𝐼‘(◡𝐼‘(𝑁‘{𝑌})))) = ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌}))) |
36 | 5, 8, 11 | dvhlmod 40578 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
37 | 12 | snssd 4809 | . . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉) |
38 | 21 | snssd 4809 | . . . . 5 ⊢ (𝜑 → {𝑌} ⊆ 𝑉) |
39 | 13, 14, 9 | lsmsp2 20966 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ {𝑋} ⊆ 𝑉 ∧ {𝑌} ⊆ 𝑉) → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌}))) |
40 | 36, 37, 38, 39 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑋})(LSSum‘𝑈)(𝑁‘{𝑌})) = (𝑁‘({𝑋} ∪ {𝑌}))) |
41 | 28, 35, 40 | 3eqtrrd 2773 | . . 3 ⊢ (𝜑 → (𝑁‘({𝑋} ∪ {𝑌})) = (𝐼‘((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌}))))) |
42 | 2, 41 | eqtrid 2780 | . 2 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) = (𝐼‘((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌}))))) |
43 | 11 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
44 | 43 | hllatd 38831 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ Lat) |
45 | 3, 5, 10 | dihcnvcl 40739 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑌}) ∈ ran 𝐼) → (◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾)) |
46 | 11, 32, 45 | syl2anc 583 | . . . 4 ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾)) |
47 | 3, 6 | latjcl 18425 | . . . 4 ⊢ ((𝐾 ∈ Lat ∧ (◡𝐼‘(𝑁‘{𝑋})) ∈ (Base‘𝐾) ∧ (◡𝐼‘(𝑁‘{𝑌})) ∈ (Base‘𝐾)) → ((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾)) |
48 | 44, 18, 46, 47 | syl3anc 1369 | . . 3 ⊢ (𝜑 → ((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾)) |
49 | 3, 5, 10 | dihcl 40738 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌}))) ∈ (Base‘𝐾)) → (𝐼‘((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌})))) ∈ ran 𝐼) |
50 | 11, 48, 49 | syl2anc 583 | . 2 ⊢ (𝜑 → (𝐼‘((◡𝐼‘(𝑁‘{𝑋}))(join‘𝐾)(◡𝐼‘(𝑁‘{𝑌})))) ∈ ran 𝐼) |
51 | 42, 50 | eqeltrd 2829 | 1 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∪ cun 3943 ⊆ wss 3945 {csn 4625 {cpr 4627 class class class wbr 5143 ◡ccnv 5672 ran crn 5674 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 lecple 17234 0gc0g 17415 joincjn 18297 Latclat 18417 LSSumclsm 19583 LModclmod 20737 LSpanclspn 20849 Atomscatm 38730 HLchlt 38817 LHypclh 39452 DVecHcdvh 40546 DIsoHcdih 40696 |
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 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-riotaBAD 38420 |
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 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-tpos 8226 df-undef 8273 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17417 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18887 df-minusg 18888 df-sbg 18889 df-subg 19072 df-cntz 19262 df-lsm 19585 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-invr 20321 df-dvr 20334 df-drng 20620 df-lmod 20739 df-lss 20810 df-lsp 20850 df-lvec 20982 df-lsatoms 38443 df-oposet 38643 df-ol 38645 df-oml 38646 df-covers 38733 df-ats 38734 df-atl 38765 df-cvlat 38789 df-hlat 38818 df-llines 38966 df-lplanes 38967 df-lvols 38968 df-lines 38969 df-psubsp 38971 df-pmap 38972 df-padd 39264 df-lhyp 39456 df-laut 39457 df-ldil 39572 df-ltrn 39573 df-trl 39627 df-tendo 40223 df-edring 40225 df-disoa 40497 df-dvech 40547 df-dib 40607 df-dic 40641 df-dih 40697 |
This theorem is referenced by: (None) |
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