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Mirrors > Home > MPE Home > Th. List > Mathboxes > cdlemn7 | Structured version Visualization version GIF version |
Description: Part of proof of Lemma N of [Crawley] p. 121 line 36. (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) |
cdlemn8.g | ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅) |
Ref | Expression |
---|---|
cdlemn7 | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝐺 = ((𝑠‘𝐹) ∘ 𝑔) ∧ ( I ↾ 𝑇) = 𝑠)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp33 1092 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉))) → 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉)) | |
2 | simp1 1054 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
3 | simp2l 1080 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | |
4 | simp2r 1081 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) | |
5 | simp31 1090 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉))) → 𝑠 ∈ 𝐸) | |
6 | simp32 1091 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉))) → 𝑔 ∈ 𝑇) | |
7 | cdlemn8.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
8 | cdlemn8.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
9 | cdlemn8.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
10 | cdlemn8.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
11 | cdlemn8.p | . . . . 5 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
12 | cdlemn8.o | . . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
13 | cdlemn8.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
14 | cdlemn8.e | . . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
15 | cdlemn8.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
16 | cdlemn8.s | . . . . 5 ⊢ + = (+g‘𝑈) | |
17 | cdlemn8.f | . . . . 5 ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄) | |
18 | 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | cdlemn6 35509 | . . . 4 ⊢ (((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〉) |
19 | 2, 3, 4, 5, 6, 18 | syl122anc 1327 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉))) → (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉) = 〈((𝑠‘𝐹) ∘ 𝑔), 𝑠〉) |
20 | 1, 19 | eqtrd 2644 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉))) → 〈𝐺, ( I ↾ 𝑇)〉 = 〈((𝑠‘𝐹) ∘ 𝑔), 𝑠〉) |
21 | fvex 6113 | . . . 4 ⊢ (𝑠‘𝐹) ∈ V | |
22 | vex 3176 | . . . 4 ⊢ 𝑔 ∈ V | |
23 | 21, 22 | coex 7011 | . . 3 ⊢ ((𝑠‘𝐹) ∘ 𝑔) ∈ V |
24 | vex 3176 | . . 3 ⊢ 𝑠 ∈ V | |
25 | 23, 24 | opth2 4875 | . 2 ⊢ (〈𝐺, ( I ↾ 𝑇)〉 = 〈((𝑠‘𝐹) ∘ 𝑔), 𝑠〉 ↔ (𝐺 = ((𝑠‘𝐹) ∘ 𝑔) ∧ ( I ↾ 𝑇) = 𝑠)) |
26 | 20, 25 | sylib 207 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝐺, ( I ↾ 𝑇)〉 = (〈(𝑠‘𝐹), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝐺 = ((𝑠‘𝐹) ∘ 𝑔) ∧ ( I ↾ 𝑇) = 𝑠)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 ↦ cmpt 4643 I cid 4948 ↾ cres 5040 ∘ ccom 5042 ‘cfv 5804 ℩crio 6510 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 lecple 15775 occoc 15776 Atomscatm 33568 HLchlt 33655 LHypclh 34288 LTrncltrn 34405 TEndoctendo 35058 DVecHcdvh 35385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-riotaBAD 33257 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-undef 7286 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-glb 16798 df-join 16799 df-meet 16800 df-p0 16862 df-p1 16863 df-lat 16869 df-clat 16931 df-oposet 33481 df-ol 33483 df-oml 33484 df-covers 33571 df-ats 33572 df-atl 33603 df-cvlat 33627 df-hlat 33656 df-llines 33802 df-lplanes 33803 df-lvols 33804 df-lines 33805 df-psubsp 33807 df-pmap 33808 df-padd 34100 df-lhyp 34292 df-laut 34293 df-ldil 34408 df-ltrn 34409 df-trl 34464 df-tendo 35061 df-edring 35063 df-dvech 35386 |
This theorem is referenced by: cdlemn8 35511 |
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