![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
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 3956 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
21 | 4, 6, 15 | dvhlvec 40712 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LVec) |
22 | mapdhe4.y | . . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
23 | 18 | eldifad 3956 | . . . . . . . . 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 21033 | . . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍}))) |
27 | 26 | simpld 493 | . . . . . . 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 41330 | . . . . . 6 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
29 | 1, 28 | eqeltrrd 2826 | . . . . 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 41331 | . . . 4 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)})))) |
31 | 1, 30 | mpbid 231 | . . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)}))) |
32 | 31 | simpld 493 | . 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 41334 | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 − 𝑍)})) = (𝐽‘{(𝐺𝑅𝐸)})) |
35 | 22 | eldifad 3956 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
36 | 7, 9, 10, 21, 35, 19, 23, 24, 25 | lspindp2 21035 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌}))) |
37 | 36 | simpld 493 | . . . . 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 41330 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) |
39 | 33, 38 | eqeltrrd 2826 | . . 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 41331 | . . . . 5 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))) |
41 | 33, 40 | mpbid 231 | . . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) |
42 | 41 | simpld 493 | . . 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 41331 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑌, 𝐺, 𝑍〉) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑌 − 𝑍)})) = (𝐽‘{(𝐺𝑅𝐸)})))) |
44 | 32, 34, 43 | mpbir2and 711 | 1 ⊢ (𝜑 → (𝐼‘〈𝑌, 𝐺, 𝑍〉) = 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 Vcvv 3461 ∖ cdif 3941 ifcif 4530 {csn 4630 {cpr 4632 〈cotp 4638 ↦ cmpt 5232 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 1st c1st 7992 2nd c2nd 7993 Basecbs 17183 0gc0g 17424 -gcsg 18900 LSpanclspn 20867 HLchlt 38952 LHypclh 39587 DVecHcdvh 40681 LCDualclcd 41189 mapdcmpd 41227 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-riotaBAD 38555 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-0g 17426 df-mre 17569 df-mrc 17570 df-acs 17572 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cntz 19280 df-oppg 19309 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lvec 21000 df-lsatoms 38578 df-lshyp 38579 df-lcv 38621 df-lfl 38660 df-lkr 38688 df-ldual 38726 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 df-tgrp 40346 df-tendo 40358 df-edring 40360 df-dveca 40606 df-disoa 40632 df-dvech 40682 df-dib 40742 df-dic 40776 df-dih 40832 df-doch 40951 df-djh 40998 df-lcdual 41190 df-mapd 41228 |
This theorem is referenced by: mapdh7dN 41353 mapdh75d 41357 mapdh8a 41378 hdmap1eq4N 41409 |
Copyright terms: Public domain | W3C validator |