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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6N | Structured version Visualization version GIF version |
Description: Part (6) of [Baer] p. 47 line 6. Note that we use ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}) which is equivalent to Baer's "Fx ∩ (Fy + Fz)" by lspdisjb 19338. TODO: If DV conditions with 𝐼 and 𝜑 become a problem later, use cbv* theorems on 𝐼 variables here to get rid of them. Maybe reorder hypotheses in lemmas to the more consistent order of this theorem, so they can be shared with this theorem. TODO: may be deleted (with its lemmas), if not needed, in view of hdmap1l6 37627. (Contributed by NM, 1-May-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdh6.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdh6.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdh6.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdh6.p | ⊢ + = (+g‘𝑈) |
mapdh6.s | ⊢ − = (-g‘𝑈) |
mapdh6.o | ⊢ 0 = (0g‘𝑈) |
mapdh6.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdh6.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdh6.d | ⊢ 𝐷 = (Base‘𝐶) |
mapdh6.a | ⊢ ✚ = (+g‘𝐶) |
mapdh6.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdh6.q | ⊢ 𝑄 = (0g‘𝐶) |
mapdh6.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdh6.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdh6.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)}))))) |
mapdh6.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdh6.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
mapdh6.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdh6.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
mapdh6.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
mapdh6.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
mapdh6.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
Ref | Expression |
---|---|
mapdh6N | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh6.q | . 2 ⊢ 𝑄 = (0g‘𝐶) | |
2 | mapdh6.i | . 2 ⊢ 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)}))))) | |
3 | mapdh6.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | mapdh6.m | . 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
5 | mapdh6.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | mapdh6.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
7 | mapdh6.s | . 2 ⊢ − = (-g‘𝑈) | |
8 | mapdh6.o | . 2 ⊢ 0 = (0g‘𝑈) | |
9 | mapdh6.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
10 | mapdh6.c | . 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
11 | mapdh6.d | . 2 ⊢ 𝐷 = (Base‘𝐶) | |
12 | mapdh6.r | . 2 ⊢ 𝑅 = (-g‘𝐶) | |
13 | mapdh6.j | . 2 ⊢ 𝐽 = (LSpan‘𝐶) | |
14 | mapdh6.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | mapdh6.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
16 | mapdh6.mn | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
17 | mapdh6.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
18 | mapdh6.p | . 2 ⊢ + = (+g‘𝑈) | |
19 | mapdh6.a | . 2 ⊢ ✚ = (+g‘𝐶) | |
20 | mapdh6.y | . 2 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
21 | mapdh6.z | . 2 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
22 | mapdh6.xn | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
23 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22 | mapdh6kN 37552 | 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 382 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∖ cdif 3720 ifcif 4225 {csn 4316 {cpr 4318 〈cotp 4324 ↦ cmpt 4863 ‘cfv 6031 ℩crio 6752 (class class class)co 6792 1st c1st 7312 2nd c2nd 7313 Basecbs 16063 +gcplusg 16148 0gc0g 16307 -gcsg 17631 LSpanclspn 19183 HLchlt 35155 LHypclh 35788 DVecHcdvh 36884 LCDualclcd 37392 mapdcmpd 37430 |
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-fal 1637 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-ot 4325 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-of 7043 df-om 7212 df-1st 7314 df-2nd 7315 df-tpos 7503 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-sets 16070 df-ress 16071 df-plusg 16161 df-mulr 16162 df-sca 16164 df-vsca 16165 df-0g 16309 df-mre 16453 df-mrc 16454 df-acs 16456 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-mgm 17449 df-sgrp 17491 df-mnd 17502 df-submnd 17543 df-grp 17632 df-minusg 17633 df-sbg 17634 df-subg 17798 df-cntz 17956 df-oppg 17982 df-lsm 18257 df-cmn 18401 df-abl 18402 df-mgp 18697 df-ur 18709 df-ring 18756 df-oppr 18830 df-dvdsr 18848 df-unit 18849 df-invr 18879 df-dvr 18890 df-drng 18958 df-lmod 19074 df-lss 19142 df-lsp 19184 df-lvec 19315 df-lsatoms 34781 df-lshyp 34782 df-lcv 34824 df-lfl 34863 df-lkr 34891 df-ldual 34929 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-tgrp 36548 df-tendo 36560 df-edring 36562 df-dveca 36808 df-disoa 36835 df-dvech 36885 df-dib 36945 df-dic 36979 df-dih 37035 df-doch 37154 df-djh 37201 df-lcdual 37393 df-mapd 37431 |
This theorem is referenced by: (None) |
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