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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1l6e | Structured version Visualization version GIF version |
Description: Lemmma for hdmap1l6 37111. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) |
Ref | Expression |
---|---|
hdmap1l6.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap1l6.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap1l6.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap1l6.p | ⊢ + = (+g‘𝑈) |
hdmap1l6.s | ⊢ − = (-g‘𝑈) |
hdmap1l6c.o | ⊢ 0 = (0g‘𝑈) |
hdmap1l6.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap1l6.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap1l6.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmap1l6.a | ⊢ ✚ = (+g‘𝐶) |
hdmap1l6.r | ⊢ 𝑅 = (-g‘𝐶) |
hdmap1l6.q | ⊢ 𝑄 = (0g‘𝐶) |
hdmap1l6.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap1l6.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap1l6.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmap1l6.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap1l6.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
hdmap1l6cl.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) |
hdmap1l6d.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
hdmap1l6d.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
hdmap1l6d.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6d.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6d.w | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6d.wn | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
Ref | Expression |
---|---|
hdmap1l6e | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap1l6.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap1l6.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap1l6.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap1l6.p | . 2 ⊢ + = (+g‘𝑈) | |
5 | hdmap1l6.s | . 2 ⊢ − = (-g‘𝑈) | |
6 | hdmap1l6c.o | . 2 ⊢ 0 = (0g‘𝑈) | |
7 | hdmap1l6.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
8 | hdmap1l6.c | . 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
9 | hdmap1l6.d | . 2 ⊢ 𝐷 = (Base‘𝐶) | |
10 | hdmap1l6.a | . 2 ⊢ ✚ = (+g‘𝐶) | |
11 | hdmap1l6.r | . 2 ⊢ 𝑅 = (-g‘𝐶) | |
12 | hdmap1l6.q | . 2 ⊢ 𝑄 = (0g‘𝐶) | |
13 | hdmap1l6.l | . 2 ⊢ 𝐿 = (LSpan‘𝐶) | |
14 | hdmap1l6.m | . 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
15 | hdmap1l6.i | . 2 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
16 | hdmap1l6.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | hdmap1l6.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
18 | hdmap1l6cl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
19 | hdmap1l6.mn | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) | |
20 | 1, 2, 16 | dvhlmod 36399 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
21 | hdmap1l6d.w | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
22 | 21 | eldifad 3586 | . . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
23 | hdmap1l6d.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
24 | 23 | eldifad 3586 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
25 | 3, 4 | lmodvacl 18877 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) ∈ 𝑉) |
26 | 20, 22, 24, 25 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ 𝑉) |
27 | 1, 2, 16 | dvhlvec 36398 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
28 | 18 | eldifad 3586 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
29 | hdmap1l6d.wn | . . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
30 | 3, 7, 27, 22, 28, 24, 29 | lspindpi 19132 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
31 | 30 | simprd 479 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})) |
32 | 3, 4, 6, 7, 20, 22, 24, 31 | lmodindp1 19014 | . . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ≠ 0 ) |
33 | eldifsn 4317 | . . 3 ⊢ ((𝑤 + 𝑌) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑤 + 𝑌) ∈ 𝑉 ∧ (𝑤 + 𝑌) ≠ 0 )) | |
34 | 26, 32, 33 | sylanbrc 698 | . 2 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ (𝑉 ∖ { 0 })) |
35 | hdmap1l6d.z | . 2 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
36 | 35 | eldifad 3586 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
37 | hdmap1l6d.yz | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
38 | hdmap1l6d.xn | . . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
39 | 3, 7, 27, 28, 24, 36, 38 | lspindpi 19132 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
40 | 39 | simpld 475 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
41 | 3, 4, 6, 7, 27, 18, 23, 35, 21, 37, 40, 29 | mapdindp3 37011 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
42 | 3, 4, 6, 7, 27, 18, 23, 35, 21, 37, 40, 29 | mapdindp4 37012 | . . . . 5 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)})) |
43 | 3, 6, 7, 27, 18, 26, 36, 41, 42 | lspindp1 19133 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)}))) |
44 | 43 | simprd 479 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)})) |
45 | prcom 4267 | . . . . 5 ⊢ {(𝑤 + 𝑌), 𝑍} = {𝑍, (𝑤 + 𝑌)} | |
46 | 45 | fveq2i 6194 | . . . 4 ⊢ (𝑁‘{(𝑤 + 𝑌), 𝑍}) = (𝑁‘{𝑍, (𝑤 + 𝑌)}) |
47 | 46 | eleq2i 2693 | . . 3 ⊢ (𝑋 ∈ (𝑁‘{(𝑤 + 𝑌), 𝑍}) ↔ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)})) |
48 | 44, 47 | sylnibr 319 | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{(𝑤 + 𝑌), 𝑍})) |
49 | 3, 7, 27, 36, 28, 26, 42 | lspindpi 19132 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}))) |
50 | 49 | simprd 479 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
51 | 50 | necomd 2849 | . 2 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) ≠ (𝑁‘{𝑍})) |
52 | eqidd 2623 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉) = (𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉)) | |
53 | eqidd 2623 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) | |
54 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 34, 35, 48, 51, 52, 53 | hdmap1l6a 37099 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 {csn 4177 {cpr 4179 〈cotp 4185 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 0gc0g 16100 -gcsg 17424 LModclmod 18863 LSpanclspn 18971 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 LCDualclcd 36875 mapdcmpd 36913 HDMap1chdma1 37081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914 df-hdmap1 37083 |
This theorem is referenced by: hdmap1l6g 37106 |
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