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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihordlem6 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.) |
Ref | Expression |
---|---|
dihordlem8.b | ⊢ 𝐵 = (Base‘𝐾) |
dihordlem8.l | ⊢ ≤ = (le‘𝐾) |
dihordlem8.a | ⊢ 𝐴 = (Atoms‘𝐾) |
dihordlem8.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dihordlem8.p | ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) |
dihordlem8.o | ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) |
dihordlem8.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
dihordlem8.e | ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) |
dihordlem8.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
dihordlem8.s | ⊢ + = (+g‘𝑈) |
dihordlem8.g | ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅) |
Ref | Expression |
---|---|
dihordlem6 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉) = 〈((𝑠‘𝐺) ∘ 𝑔), 𝑠〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1138 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | simp2r 1202 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) | |
3 | simp2l 1201 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | |
4 | simp3 1140 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) | |
5 | dihordlem8.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
6 | dihordlem8.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
7 | dihordlem8.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
8 | dihordlem8.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
9 | dihordlem8.p | . . 3 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
10 | dihordlem8.o | . . 3 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
11 | dihordlem8.t | . . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
12 | dihordlem8.e | . . 3 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
13 | dihordlem8.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
14 | dihordlem8.s | . . 3 ⊢ + = (+g‘𝑈) | |
15 | dihordlem8.g | . . 3 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅) | |
16 | 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 | cdlemn6 38958 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉) = 〈((𝑠‘𝐺) ∘ 𝑔), 𝑠〉) |
17 | 1, 2, 3, 4, 16 | syl121anc 1377 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇)) → (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉) = 〈((𝑠‘𝐺) ∘ 𝑔), 𝑠〉) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 〈cop 4552 class class class wbr 5058 ↦ cmpt 5140 I cid 5459 ↾ cres 5558 ∘ ccom 5560 ‘cfv 6385 ℩crio 7174 (class class class)co 7218 Basecbs 16765 +gcplusg 16807 lecple 16814 occoc 16815 Atomscatm 37019 HLchlt 37106 LHypclh 37740 LTrncltrn 37857 TEndoctendo 38508 DVecHcdvh 38834 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5184 ax-sep 5197 ax-nul 5204 ax-pow 5263 ax-pr 5327 ax-un 7528 ax-cnex 10790 ax-resscn 10791 ax-1cn 10792 ax-icn 10793 ax-addcl 10794 ax-addrcl 10795 ax-mulcl 10796 ax-mulrcl 10797 ax-mulcom 10798 ax-addass 10799 ax-mulass 10800 ax-distr 10801 ax-i2m1 10802 ax-1ne0 10803 ax-1rid 10804 ax-rnegex 10805 ax-rrecex 10806 ax-cnre 10807 ax-pre-lttri 10808 ax-pre-lttrn 10809 ax-pre-ltadd 10810 ax-pre-mulgt0 10811 ax-riotaBAD 36709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3415 df-sbc 3700 df-csb 3817 df-dif 3874 df-un 3876 df-in 3878 df-ss 3888 df-pss 3890 df-nul 4243 df-if 4445 df-pw 4520 df-sn 4547 df-pr 4549 df-tp 4551 df-op 4553 df-uni 4825 df-iun 4911 df-iin 4912 df-br 5059 df-opab 5121 df-mpt 5141 df-tr 5167 df-id 5460 df-eprel 5465 df-po 5473 df-so 5474 df-fr 5514 df-we 5516 df-xp 5562 df-rel 5563 df-cnv 5564 df-co 5565 df-dm 5566 df-rn 5567 df-res 5568 df-ima 5569 df-pred 6165 df-ord 6221 df-on 6222 df-lim 6223 df-suc 6224 df-iota 6343 df-fun 6387 df-fn 6388 df-f 6389 df-f1 6390 df-fo 6391 df-f1o 6392 df-fv 6393 df-riota 7175 df-ov 7221 df-oprab 7222 df-mpo 7223 df-om 7650 df-1st 7766 df-2nd 7767 df-undef 8020 df-wrecs 8052 df-recs 8113 df-rdg 8151 df-1o 8207 df-er 8396 df-map 8515 df-en 8632 df-dom 8633 df-sdom 8634 df-fin 8635 df-pnf 10874 df-mnf 10875 df-xr 10876 df-ltxr 10877 df-le 10878 df-sub 11069 df-neg 11070 df-nn 11836 df-2 11898 df-3 11899 df-4 11900 df-5 11901 df-6 11902 df-n0 12096 df-z 12182 df-uz 12444 df-fz 13101 df-struct 16705 df-slot 16740 df-ndx 16750 df-base 16766 df-plusg 16820 df-mulr 16821 df-sca 16823 df-vsca 16824 df-proset 17807 df-poset 17825 df-plt 17841 df-lub 17857 df-glb 17858 df-join 17859 df-meet 17860 df-p0 17936 df-p1 17937 df-lat 17943 df-clat 18010 df-oposet 36932 df-ol 36934 df-oml 36935 df-covers 37022 df-ats 37023 df-atl 37054 df-cvlat 37078 df-hlat 37107 df-llines 37254 df-lplanes 37255 df-lvols 37256 df-lines 37257 df-psubsp 37259 df-pmap 37260 df-padd 37552 df-lhyp 37744 df-laut 37745 df-ldil 37860 df-ltrn 37861 df-trl 37915 df-tendo 38511 df-edring 38513 df-dvech 38835 |
This theorem is referenced by: dihordlem7 38970 |
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