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Mirrors > Home > MPE Home > Th. List > Mathboxes > dihordlem7b | Structured version Visualization version GIF version |
Description: Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (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 |
---|---|
dihordlem7b | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝑓 = 𝑔 ∧ 𝑂 = 𝑠)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dihordlem8.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | dihordlem8.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
3 | dihordlem8.a | . . . . 5 ⊢ 𝐴 = (Atoms‘𝐾) | |
4 | dihordlem8.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | dihordlem8.p | . . . . 5 ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) | |
6 | dihordlem8.o | . . . . 5 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) | |
7 | dihordlem8.t | . . . . 5 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
8 | dihordlem8.e | . . . . 5 ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) | |
9 | dihordlem8.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
10 | dihordlem8.s | . . . . 5 ⊢ + = (+g‘𝑈) | |
11 | dihordlem8.g | . . . . 5 ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | dihordlem7 37020 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝑓 = ((𝑠‘𝐺) ∘ 𝑔) ∧ 𝑂 = 𝑠)) |
13 | 12 | simpld 476 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → 𝑓 = ((𝑠‘𝐺) ∘ 𝑔)) |
14 | 12 | simprd 477 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → 𝑂 = 𝑠) |
15 | 14 | fveq1d 6334 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝑂‘𝐺) = (𝑠‘𝐺)) |
16 | simp1 1130 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | 2, 3, 4, 5 | lhpocnel2 35823 | . . . . . . . 8 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
18 | 17 | 3ad2ant1 1127 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) |
19 | simp2r 1242 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) | |
20 | 2, 3, 4, 7, 11 | ltrniotacl 36384 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) → 𝐺 ∈ 𝑇) |
21 | 16, 18, 19, 20 | syl3anc 1476 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → 𝐺 ∈ 𝑇) |
22 | 6, 1 | tendo02 36592 | . . . . . 6 ⊢ (𝐺 ∈ 𝑇 → (𝑂‘𝐺) = ( I ↾ 𝐵)) |
23 | 21, 22 | syl 17 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝑂‘𝐺) = ( I ↾ 𝐵)) |
24 | 15, 23 | eqtr3d 2807 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝑠‘𝐺) = ( I ↾ 𝐵)) |
25 | 24 | coeq1d 5422 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → ((𝑠‘𝐺) ∘ 𝑔) = (( I ↾ 𝐵) ∘ 𝑔)) |
26 | simp32 1252 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → 𝑔 ∈ 𝑇) | |
27 | 1, 4, 7 | ltrn1o 35928 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑔 ∈ 𝑇) → 𝑔:𝐵–1-1-onto→𝐵) |
28 | 16, 26, 27 | syl2anc 565 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → 𝑔:𝐵–1-1-onto→𝐵) |
29 | f1of 6278 | . . . 4 ⊢ (𝑔:𝐵–1-1-onto→𝐵 → 𝑔:𝐵⟶𝐵) | |
30 | fcoi2 6219 | . . . 4 ⊢ (𝑔:𝐵⟶𝐵 → (( I ↾ 𝐵) ∘ 𝑔) = 𝑔) | |
31 | 28, 29, 30 | 3syl 18 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → (( I ↾ 𝐵) ∘ 𝑔) = 𝑔) |
32 | 13, 25, 31 | 3eqtrd 2809 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → 𝑓 = 𝑔) |
33 | 32, 14 | jca 495 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ (𝑠 ∈ 𝐸 ∧ 𝑔 ∈ 𝑇 ∧ 〈𝑓, 𝑂〉 = (〈(𝑠‘𝐺), 𝑠〉 + 〈𝑔, 𝑂〉))) → (𝑓 = 𝑔 ∧ 𝑂 = 𝑠)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 ∧ w3a 1071 = wceq 1631 ∈ wcel 2145 〈cop 4322 class class class wbr 4786 ↦ cmpt 4863 I cid 5156 ↾ cres 5251 ∘ ccom 5253 ⟶wf 6027 –1-1-onto→wf1o 6030 ‘cfv 6031 ℩crio 6752 (class class class)co 6792 Basecbs 16063 +gcplusg 16148 lecple 16155 occoc 16156 Atomscatm 35068 HLchlt 35155 LHypclh 35788 LTrncltrn 35905 TEndoctendo 36557 DVecHcdvh 36884 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7095 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-riotaBAD 34757 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-undef 7550 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-n0 11494 df-z 11579 df-uz 11888 df-fz 12533 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-plusg 16161 df-mulr 16162 df-sca 16164 df-vsca 16165 df-preset 17135 df-poset 17153 df-plt 17165 df-lub 17181 df-glb 17182 df-join 17183 df-meet 17184 df-p0 17246 df-p1 17247 df-lat 17253 df-clat 17315 df-oposet 34981 df-ol 34983 df-oml 34984 df-covers 35071 df-ats 35072 df-atl 35103 df-cvlat 35127 df-hlat 35156 df-llines 35302 df-lplanes 35303 df-lvols 35304 df-lines 35305 df-psubsp 35307 df-pmap 35308 df-padd 35600 df-lhyp 35792 df-laut 35793 df-ldil 35908 df-ltrn 35909 df-trl 35964 df-tendo 36560 df-edring 36562 df-dvech 36885 |
This theorem is referenced by: dihord10 37029 |
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