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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdheq4 | Structured version Visualization version GIF version |
Description: Lemma for ~? mapdh . Part (4) in [Baer] p. 46. (Contributed by NM, 12-Apr-2015.) |
Ref | Expression |
---|---|
mapdh.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh.i | ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) |
mapdh.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdh.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdh.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdh.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdh.s | ⊢ − = (-g‘𝑈) |
mapdhc.o | ⊢ 0 = (0g‘𝑈) |
mapdh.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdh.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdh.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdh.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdh.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdh.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdhc.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
mapdhcl.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdhe4.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdhe.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdh.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
mapdh.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) |
mapdh.eg | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) |
mapdh.ee | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) |
Ref | Expression |
---|---|
mapdheq4 | ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑍〉) = 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.ee | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸) | |
2 | mapdh.q | . . . . 5 ⊢ 𝑄 = (0g‘𝐶) | |
3 | mapdh.i | . . . . 5 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1st ‘(1st ‘𝑥)) − (2nd ‘𝑥))})) = (𝐽‘{((2nd ‘(1st ‘𝑥))𝑅ℎ)}))))) | |
4 | mapdh.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | mapdh.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
6 | mapdh.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
7 | mapdh.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
8 | mapdh.s | . . . . 5 ⊢ − = (-g‘𝑈) | |
9 | mapdhc.o | . . . . 5 ⊢ 0 = (0g‘𝑈) | |
10 | mapdh.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
11 | mapdh.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
12 | mapdh.d | . . . . 5 ⊢ 𝐷 = (Base‘𝐶) | |
13 | mapdh.r | . . . . 5 ⊢ 𝑅 = (-g‘𝐶) | |
14 | mapdh.j | . . . . 5 ⊢ 𝐽 = (LSpan‘𝐶) | |
15 | mapdh.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
16 | mapdhc.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
17 | mapdh.mn | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
18 | mapdhcl.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
19 | mapdhe.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
20 | 19 | eldifad 3619 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
21 | 4, 6, 15 | dvhlvec 36715 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LVec) |
22 | mapdhe4.y | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
23 | 18 | eldifad 3619 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
24 | mapdh.yz | . . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) | |
25 | mapdh.xn | . . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
26 | 7, 9, 10, 21, 22, 20, 23, 24, 25 | lspindp1 19181 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))) |
27 | 26 | simpld 474 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
28 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 27 | mapdhcl 37333 | . . . . . 6 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
29 | 1, 28 | eqeltrrd 2731 | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐷) |
30 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 29, 27 | mapdheq 37334 | . . . 4 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)})))) |
31 | 1, 30 | mpbid 222 | . . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)}))) |
32 | 31 | simpld 474 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸})) |
33 | mapdh.eg | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺) | |
34 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 19, 25, 24, 33, 1 | mapdheq4lem 37337 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 − 𝑍)})) = (𝐽‘{(𝐺𝑅𝐸)})) |
35 | 22 | eldifad 3619 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
36 | 7, 9, 10, 21, 35, 19, 23, 24, 25 | lspindp2 19183 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))) |
37 | 36 | simpld 474 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
38 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 35, 37 | mapdhcl 37333 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) |
39 | 33, 38 | eqeltrrd 2731 | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷) |
40 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 39, 37 | mapdheq 37334 | . . . . 5 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))) |
41 | 33, 40 | mpbid 222 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) |
42 | 41 | simpld 474 | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})) |
43 | 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 42, 22, 19, 29, 24 | mapdheq 37334 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑌, 𝐺, 𝑍〉) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑌 − 𝑍)})) = (𝐽‘{(𝐺𝑅𝐸)})))) |
44 | 32, 34, 43 | mpbir2and 977 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑍〉) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 Vcvv 3231 ∖ cdif 3604 ifcif 4119 {csn 4210 {cpr 4212 〈cotp 4218 ↦ cmpt 4762 ‘cfv 5926 ℩crio 6650 (class class class)co 6690 1st c1st 7208 2nd c2nd 7209 Basecbs 15904 0gc0g 16147 -gcsg 17471 LSpanclspn 19019 HLchlt 34955 LHypclh 35588 DVecHcdvh 36684 LCDualclcd 37192 mapdcmpd 37230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-riotaBAD 34557 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-ot 4219 df-uni 4469 df-int 4508 df-iun 4554 df-iin 4555 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-of 6939 df-om 7108 df-1st 7210 df-2nd 7211 df-tpos 7397 df-undef 7444 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-n0 11331 df-z 11416 df-uz 11726 df-fz 12365 df-struct 15906 df-ndx 15907 df-slot 15908 df-base 15910 df-sets 15911 df-ress 15912 df-plusg 16001 df-mulr 16002 df-sca 16004 df-vsca 16005 df-0g 16149 df-mre 16293 df-mrc 16294 df-acs 16296 df-preset 16975 df-poset 16993 df-plt 17005 df-lub 17021 df-glb 17022 df-join 17023 df-meet 17024 df-p0 17086 df-p1 17087 df-lat 17093 df-clat 17155 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-submnd 17383 df-grp 17472 df-minusg 17473 df-sbg 17474 df-subg 17638 df-cntz 17796 df-oppg 17822 df-lsm 18097 df-cmn 18241 df-abl 18242 df-mgp 18536 df-ur 18548 df-ring 18595 df-oppr 18669 df-dvdsr 18687 df-unit 18688 df-invr 18718 df-dvr 18729 df-drng 18797 df-lmod 18913 df-lss 18981 df-lsp 19020 df-lvec 19151 df-lsatoms 34581 df-lshyp 34582 df-lcv 34624 df-lfl 34663 df-lkr 34691 df-ldual 34729 df-oposet 34781 df-ol 34783 df-oml 34784 df-covers 34871 df-ats 34872 df-atl 34903 df-cvlat 34927 df-hlat 34956 df-llines 35102 df-lplanes 35103 df-lvols 35104 df-lines 35105 df-psubsp 35107 df-pmap 35108 df-padd 35400 df-lhyp 35592 df-laut 35593 df-ldil 35708 df-ltrn 35709 df-trl 35764 df-tgrp 36348 df-tendo 36360 df-edring 36362 df-dveca 36608 df-disoa 36635 df-dvech 36685 df-dib 36745 df-dic 36779 df-dih 36835 df-doch 36954 df-djh 37001 df-lcdual 37193 df-mapd 37231 |
This theorem is referenced by: mapdh7dN 37356 mapdh75d 37360 mapdh8a 37381 hdmap1eq4N 37413 |
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