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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 1213 | . . 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 1138 | . . . 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 1201 | . . . 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 1202 | . . . 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 1211 | . . . 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 1212 | . . . 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 38902 | . . . 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 1381 | . . 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 2771 | . 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 6708 | . . . 4 ⊢ (𝑠‘𝐹) ∈ V | |
22 | vex 3402 | . . . 4 ⊢ 𝑔 ∈ V | |
23 | 21, 22 | coex 7686 | . . 3 ⊢ ((𝑠‘𝐹) ∘ 𝑔) ∈ V |
24 | vex 3402 | . . 3 ⊢ 𝑠 ∈ V | |
25 | 23, 24 | opth2 5349 | . 2 ⊢ (〈𝐺, ( I ↾ 𝑇)〉 = 〈((𝑠‘𝐹) ∘ 𝑔), 𝑠〉 ↔ (𝐺 = ((𝑠‘𝐹) ∘ 𝑔) ∧ ( I ↾ 𝑇) = 𝑠)) |
26 | 20, 25 | sylib 221 | 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 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 〈cop 4533 class class class wbr 5039 ↦ cmpt 5120 I cid 5439 ↾ cres 5538 ∘ ccom 5540 ‘cfv 6358 ℩crio 7147 (class class class)co 7191 Basecbs 16666 +gcplusg 16749 lecple 16756 occoc 16757 Atomscatm 36963 HLchlt 37050 LHypclh 37684 LTrncltrn 37801 TEndoctendo 38452 DVecHcdvh 38778 |
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 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-riotaBAD 36653 |
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 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-undef 7993 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-n0 12056 df-z 12142 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-plusg 16762 df-mulr 16763 df-sca 16765 df-vsca 16766 df-proset 17756 df-poset 17774 df-plt 17790 df-lub 17806 df-glb 17807 df-join 17808 df-meet 17809 df-p0 17885 df-p1 17886 df-lat 17892 df-clat 17959 df-oposet 36876 df-ol 36878 df-oml 36879 df-covers 36966 df-ats 36967 df-atl 36998 df-cvlat 37022 df-hlat 37051 df-llines 37198 df-lplanes 37199 df-lvols 37200 df-lines 37201 df-psubsp 37203 df-pmap 37204 df-padd 37496 df-lhyp 37688 df-laut 37689 df-ldil 37804 df-ltrn 37805 df-trl 37859 df-tendo 38455 df-edring 38457 df-dvech 38779 |
This theorem is referenced by: cdlemn8 38904 |
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