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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemn6 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.) |
Ref | Expression |
---|---|
cdlemn8.b | ⊢ 𝐵 = (Base‘𝐾) |
cdlemn8.l | ⊢ ≤ = (le‘𝐾) |
cdlemn8.a | ⊢ 𝐴 = (Atoms‘𝐾) |
cdlemn8.h | ⊢ 𝐻 = (LHyp‘𝐾) |
cdlemn8.p | ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
cdlemn8.o | ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
cdlemn8.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
cdlemn8.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
cdlemn8.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
cdlemn8.s | ⊢ + = (+g‘𝑈) |
cdlemn8.f | ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄) |
Ref | Expression |
---|---|
cdlemn6 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉) = 〈((𝑠‘𝐹) ∘ 𝑔), 𝑠〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1061 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simp3l 1089 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → 𝑠 ∈ 𝐸) | |
3 | cdlemn8.l | . . . . . . 7 ⊢ ≤ = (le‘𝐾) | |
4 | cdlemn8.a | . . . . . . 7 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | cdlemn8.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
6 | cdlemn8.p | . . . . . . 7 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
7 | 3, 4, 5, 6 | lhpocnel2 35305 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
9 | simp2l 1087 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | |
10 | cdlemn8.t | . . . . . 6 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
11 | cdlemn8.f | . . . . . 6 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄) | |
12 | 3, 4, 5, 10, 11 | ltrniotacl 35867 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
13 | 1, 8, 9, 12 | syl3anc 1326 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → 𝐹 ∈ 𝑇) |
14 | cdlemn8.e | . . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
15 | 5, 10, 14 | tendocl 36055 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑠‘𝐹) ∈ 𝑇) |
16 | 1, 2, 13, 15 | syl3anc 1326 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠‘𝐹) ∈ 𝑇) |
17 | simp3r 1090 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → 𝑔 ∈ 𝑇) | |
18 | cdlemn8.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
19 | cdlemn8.o | . . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
20 | 18, 5, 10, 14, 19 | tendo0cl 36078 | . . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑂 ∈ 𝐸) |
21 | 1, 20 | syl 17 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → 𝑂 ∈ 𝐸) |
22 | cdlemn8.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
23 | eqid 2622 | . . . 4 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
24 | cdlemn8.s | . . . 4 ⊢ + = (+g‘𝑈) | |
25 | eqid 2622 | . . . 4 ⊢ (+g‘(Scalar‘𝑈)) = (+g‘(Scalar‘𝑈)) | |
26 | 5, 10, 14, 22, 23, 24, 25 | dvhopvadd 36382 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑠‘𝐹) ∈ 𝑇 ∧ 𝑠 ∈ 𝐸) ∧ (𝑔 ∈ 𝑇 ∧ 𝑂 ∈ 𝐸)) → (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉) = 〈((𝑠‘𝐹) ∘ 𝑔), (𝑠(+g‘(Scalar‘𝑈))𝑂)〉) |
27 | 1, 16, 2, 17, 21, 26 | syl122anc 1335 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉) = 〈((𝑠‘𝐹) ∘ 𝑔), (𝑠(+g‘(Scalar‘𝑈))𝑂)〉) |
28 | eqid 2622 | . . . . . . 7 ⊢ (𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ)))) = (𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ)))) | |
29 | 5, 10, 14, 22, 23, 28, 25 | dvhfplusr 36373 | . . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (+g‘(Scalar‘𝑈)) = (𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ))))) |
30 | 1, 29 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (+g‘(Scalar‘𝑈)) = (𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ))))) |
31 | 30 | oveqd 6667 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠(+g‘(Scalar‘𝑈))𝑂) = (𝑠(𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ))))𝑂)) |
32 | 18, 5, 10, 14, 19, 28 | tendo0plr 36080 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑠 ∈ 𝐸) → (𝑠(𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ))))𝑂) = 𝑠) |
33 | 1, 2, 32 | syl2anc 693 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠(𝑡 ∈ 𝐸, 𝑢 ∈ 𝐸 ↦ (ℎ ∈ 𝑇 ↦ ((𝑡‘ℎ) ∘ (𝑢‘ℎ))))𝑂) = 𝑠) |
34 | 31, 33 | eqtrd 2656 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠(+g‘(Scalar‘𝑈))𝑂) = 𝑠) |
35 | 34 | opeq2d 4409 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → 〈((𝑠‘𝐹) ∘ 𝑔), (𝑠(+g‘(Scalar‘𝑈))𝑂)〉 = 〈((𝑠‘𝐹) ∘ 𝑔), 𝑠〉) |
36 | 27, 35 | eqtrd 2656 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉) = 〈((𝑠‘𝐹) ∘ 𝑔), 𝑠〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 〈cop 4183 class class class wbr 4653 ↦ cmpt 4729 I cid 5023 ↾ cres 5116 ∘ ccom 5118 ‘cfv 5888 ℩crio 6610 (class class class)co 6650 ↦ cmpt2 6652 Basecbs 15857 +gcplusg 15941 Scalarcsca 15944 lecple 15948 occoc 15949 Atomscatm 34550 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 TEndoctendo 36040 DVecHcdvh 36367 |
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-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-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-om 7066 df-1st 7168 df-2nd 7169 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-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 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-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-tendo 36043 df-edring 36045 df-dvech 36368 |
This theorem is referenced by: cdlemn7 36492 dihordlem6 36502 |
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