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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6iN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 37556. Eliminate auxiliary vector 𝑤. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
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 })) |
mapdh.p | ⊢ + = (+g‘𝑈) |
mapdh.a | ⊢ ✚ = (+g‘𝐶) |
mapdh6i.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
mapdh6i.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdh6i.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdh6i.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
Ref | Expression |
---|---|
mapdh6iN | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdh.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | mapdh.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
4 | mapdh.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
5 | mapdh.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | mapdhcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
7 | 6 | eldifad 3727 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
8 | mapdh6i.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
9 | 8 | eldifad 3727 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
10 | 1, 2, 3, 4, 5, 7, 9 | dvh3dim 37255 | . 2 ⊢ (𝜑 → ∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
11 | mapdh.q | . . . 4 ⊢ 𝑄 = (0g‘𝐶) | |
12 | mapdh.i | . . . 4 ⊢ 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)}))))) | |
13 | mapdh.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
14 | mapdh.s | . . . 4 ⊢ − = (-g‘𝑈) | |
15 | mapdhc.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
16 | mapdh.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
17 | mapdh.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
18 | mapdh.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
19 | mapdh.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
20 | 5 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
21 | mapdhc.f | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
22 | 21 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝐹 ∈ 𝐷) |
23 | mapdh.mn | . . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
24 | 23 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
25 | 6 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
26 | mapdh.p | . . . 4 ⊢ + = (+g‘𝑈) | |
27 | mapdh.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
28 | mapdh6i.xn | . . . . 5 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
29 | 28 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
30 | mapdh6i.yz | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
31 | 30 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
32 | 8 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
33 | mapdh6i.z | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
34 | 33 | 3ad2ant1 1128 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
35 | eqid 2760 | . . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈) | |
36 | 1, 2, 5 | dvhlmod 36919 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod) |
37 | 36 | 3ad2ant1 1128 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑈 ∈ LMod) |
38 | 3, 35, 4, 36, 7, 9 | lspprcl 19200 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
39 | 38 | 3ad2ant1 1128 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝑁‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈)) |
40 | simp2 1132 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ 𝑉) | |
41 | simp3 1133 | . . . . 5 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
42 | 3, 15, 35, 37, 39, 40, 41 | lssneln0 19174 | . . . 4 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → 𝑤 ∈ (𝑉 ∖ { 0 })) |
43 | 11, 12, 1, 13, 2, 3, 14, 15, 4, 16, 17, 18, 19, 20, 22, 24, 25, 26, 27, 29, 31, 32, 34, 42, 41 | mapdh6hN 37552 | . . 3 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉 ∧ ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
44 | 43 | rexlimdv3a 3171 | . 2 ⊢ (𝜑 → (∃𝑤 ∈ 𝑉 ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌}) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
45 | 10, 44 | mpd 15 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1072 = wceq 1632 ∈ wcel 2139 ∃wrex 3051 Vcvv 3340 ∖ cdif 3712 ifcif 4230 {csn 4321 {cpr 4323 〈cotp 4329 ↦ cmpt 4881 ‘cfv 6049 ℩crio 6774 (class class class)co 6814 1st c1st 7332 2nd c2nd 7333 Basecbs 16079 +gcplusg 16163 0gc0g 16322 -gcsg 17645 LModclmod 19085 LSubSpclss 19154 LSpanclspn 19193 HLchlt 35158 LHypclh 35791 DVecHcdvh 36887 LCDualclcd 37395 mapdcmpd 37433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7115 ax-cnex 10204 ax-resscn 10205 ax-1cn 10206 ax-icn 10207 ax-addcl 10208 ax-addrcl 10209 ax-mulcl 10210 ax-mulrcl 10211 ax-mulcom 10212 ax-addass 10213 ax-mulass 10214 ax-distr 10215 ax-i2m1 10216 ax-1ne0 10217 ax-1rid 10218 ax-rnegex 10219 ax-rrecex 10220 ax-cnre 10221 ax-pre-lttri 10222 ax-pre-lttrn 10223 ax-pre-ltadd 10224 ax-pre-mulgt0 10225 ax-riotaBAD 34760 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-ot 4330 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6775 df-ov 6817 df-oprab 6818 df-mpt2 6819 df-of 7063 df-om 7232 df-1st 7334 df-2nd 7335 df-tpos 7522 df-undef 7569 df-wrecs 7577 df-recs 7638 df-rdg 7676 df-1o 7730 df-oadd 7734 df-er 7913 df-map 8027 df-en 8124 df-dom 8125 df-sdom 8126 df-fin 8127 df-pnf 10288 df-mnf 10289 df-xr 10290 df-ltxr 10291 df-le 10292 df-sub 10480 df-neg 10481 df-nn 11233 df-2 11291 df-3 11292 df-4 11293 df-5 11294 df-6 11295 df-n0 11505 df-z 11590 df-uz 11900 df-fz 12540 df-struct 16081 df-ndx 16082 df-slot 16083 df-base 16085 df-sets 16086 df-ress 16087 df-plusg 16176 df-mulr 16177 df-sca 16179 df-vsca 16180 df-0g 16324 df-mre 16468 df-mrc 16469 df-acs 16471 df-preset 17149 df-poset 17167 df-plt 17179 df-lub 17195 df-glb 17196 df-join 17197 df-meet 17198 df-p0 17260 df-p1 17261 df-lat 17267 df-clat 17329 df-mgm 17463 df-sgrp 17505 df-mnd 17516 df-submnd 17557 df-grp 17646 df-minusg 17647 df-sbg 17648 df-subg 17812 df-cntz 17970 df-oppg 17996 df-lsm 18271 df-cmn 18415 df-abl 18416 df-mgp 18710 df-ur 18722 df-ring 18769 df-oppr 18843 df-dvdsr 18861 df-unit 18862 df-invr 18892 df-dvr 18903 df-drng 18971 df-lmod 19087 df-lss 19155 df-lsp 19194 df-lvec 19325 df-lsatoms 34784 df-lshyp 34785 df-lcv 34827 df-lfl 34866 df-lkr 34894 df-ldual 34932 df-oposet 34984 df-ol 34986 df-oml 34987 df-covers 35074 df-ats 35075 df-atl 35106 df-cvlat 35130 df-hlat 35159 df-llines 35305 df-lplanes 35306 df-lvols 35307 df-lines 35308 df-psubsp 35310 df-pmap 35311 df-padd 35603 df-lhyp 35795 df-laut 35796 df-ldil 35911 df-ltrn 35912 df-trl 35967 df-tgrp 36551 df-tendo 36563 df-edring 36565 df-dveca 36811 df-disoa 36838 df-dvech 36888 df-dib 36948 df-dic 36982 df-dih 37038 df-doch 37157 df-djh 37204 df-lcdual 37396 df-mapd 37434 |
This theorem is referenced by: mapdh6jN 37554 |
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