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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem4 | Structured version Visualization version GIF version |
Description: Simplify (𝐴 ∖ {𝑄}) in hdmap14lem3 40324 to provide a slightly simpler definition later. TODO: Use hdmap14lem4a 40325 if that one is also used directly elsewhere. Otherwise, merge hdmap14lem4a 40325 into this one. (Contributed by NM, 31-May-2015.) |
Ref | Expression |
---|---|
hdmap14lem1.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap14lem1.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap14lem1.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap14lem1.t | ⊢ · = ( ·𝑠 ‘𝑈) |
hdmap14lem3.o | ⊢ 0 = (0g‘𝑈) |
hdmap14lem1.r | ⊢ 𝑅 = (Scalar‘𝑈) |
hdmap14lem1.b | ⊢ 𝐵 = (Base‘𝑅) |
hdmap14lem1.z | ⊢ 𝑍 = (0g‘𝑅) |
hdmap14lem1.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap14lem2.e | ⊢ ∙ = ( ·𝑠 ‘𝐶) |
hdmap14lem1.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap14lem2.p | ⊢ 𝑃 = (Scalar‘𝐶) |
hdmap14lem2.a | ⊢ 𝐴 = (Base‘𝑃) |
hdmap14lem2.q | ⊢ 𝑄 = (0g‘𝑃) |
hdmap14lem1.s | ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) |
hdmap14lem1.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap14lem3.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap14lem1.f | ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) |
Ref | Expression |
---|---|
hdmap14lem4 | ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap14lem1.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap14lem1.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap14lem1.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap14lem1.t | . . 3 ⊢ · = ( ·𝑠 ‘𝑈) | |
5 | hdmap14lem3.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
6 | hdmap14lem1.r | . . 3 ⊢ 𝑅 = (Scalar‘𝑈) | |
7 | hdmap14lem1.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
8 | hdmap14lem1.z | . . 3 ⊢ 𝑍 = (0g‘𝑅) | |
9 | hdmap14lem1.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
10 | hdmap14lem2.e | . . 3 ⊢ ∙ = ( ·𝑠 ‘𝐶) | |
11 | hdmap14lem1.l | . . 3 ⊢ 𝐿 = (LSpan‘𝐶) | |
12 | hdmap14lem2.p | . . 3 ⊢ 𝑃 = (Scalar‘𝐶) | |
13 | hdmap14lem2.a | . . 3 ⊢ 𝐴 = (Base‘𝑃) | |
14 | hdmap14lem2.q | . . 3 ⊢ 𝑄 = (0g‘𝑃) | |
15 | hdmap14lem1.s | . . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊) | |
16 | hdmap14lem1.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | hdmap14lem3.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
18 | hdmap14lem1.f | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ∖ {𝑍})) | |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | hdmap14lem3 40324 | . 2 ⊢ (𝜑 → ∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | hdmap14lem4a 40325 | . 2 ⊢ (𝜑 → (∃!𝑔 ∈ (𝐴 ∖ {𝑄})(𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)) ↔ ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋)))) |
21 | 19, 20 | mpbid 231 | 1 ⊢ (𝜑 → ∃!𝑔 ∈ 𝐴 (𝑆‘(𝐹 · 𝑋)) = (𝑔 ∙ (𝑆‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃!wreu 3351 ∖ cdif 3907 {csn 4586 ‘cfv 6496 (class class class)co 7356 Basecbs 17082 Scalarcsca 17135 ·𝑠 cvsca 17136 0gc0g 17320 LSpanclspn 20430 HLchlt 37803 LHypclh 38438 DVecHcdvh 39532 LCDualclcd 40040 HDMapchdma 40246 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-cnex 11106 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 ax-riotaBAD 37406 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-ot 4595 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7616 df-om 7802 df-1st 7920 df-2nd 7921 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-1o 8411 df-er 8647 df-map 8766 df-en 8883 df-dom 8884 df-sdom 8885 df-fin 8886 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-nn 12153 df-2 12215 df-3 12216 df-4 12217 df-5 12218 df-6 12219 df-n0 12413 df-z 12499 df-uz 12763 df-fz 13424 df-struct 17018 df-sets 17035 df-slot 17053 df-ndx 17065 df-base 17083 df-ress 17112 df-plusg 17145 df-mulr 17146 df-sca 17148 df-vsca 17149 df-0g 17322 df-mre 17465 df-mrc 17466 df-acs 17468 df-proset 18183 df-poset 18201 df-plt 18218 df-lub 18234 df-glb 18235 df-join 18236 df-meet 18237 df-p0 18313 df-p1 18314 df-lat 18320 df-clat 18387 df-mgm 18496 df-sgrp 18545 df-mnd 18556 df-submnd 18601 df-grp 18750 df-minusg 18751 df-sbg 18752 df-subg 18923 df-cntz 19095 df-oppg 19122 df-lsm 19416 df-cmn 19562 df-abl 19563 df-mgp 19895 df-ur 19912 df-ring 19964 df-oppr 20047 df-dvdsr 20068 df-unit 20069 df-invr 20099 df-dvr 20110 df-drng 20185 df-lmod 20322 df-lss 20391 df-lsp 20431 df-lvec 20562 df-lsatoms 37429 df-lshyp 37430 df-lcv 37472 df-lfl 37511 df-lkr 37539 df-ldual 37577 df-oposet 37629 df-ol 37631 df-oml 37632 df-covers 37719 df-ats 37720 df-atl 37751 df-cvlat 37775 df-hlat 37804 df-llines 37952 df-lplanes 37953 df-lvols 37954 df-lines 37955 df-psubsp 37957 df-pmap 37958 df-padd 38250 df-lhyp 38442 df-laut 38443 df-ldil 38558 df-ltrn 38559 df-trl 38613 df-tgrp 39197 df-tendo 39209 df-edring 39211 df-dveca 39457 df-disoa 39483 df-dvech 39533 df-dib 39593 df-dic 39627 df-dih 39683 df-doch 39802 df-djh 39849 df-lcdual 40041 df-mapd 40079 df-hvmap 40211 df-hdmap1 40247 df-hdmap 40248 |
This theorem is referenced by: hdmap14lem7 40328 |
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