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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6kN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 40524. Eliminate nonzero vector requirement. (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) |
mapdh6k.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
mapdh6k.z | ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
mapdh6k.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
Ref | Expression |
---|---|
mapdh6kN | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.q | . . 3 ⊢ 𝑄 = (0g‘𝐶) | |
2 | mapdh.i | . . 3 ⊢ 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 | mapdh.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | mapdh.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
5 | mapdh.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | mapdh.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
7 | mapdh.s | . . 3 ⊢ − = (-g‘𝑈) | |
8 | mapdhc.o | . . 3 ⊢ 0 = (0g‘𝑈) | |
9 | mapdh.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
10 | mapdh.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
11 | mapdh.d | . . 3 ⊢ 𝐷 = (Base‘𝐶) | |
12 | mapdh.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
13 | mapdh.j | . . 3 ⊢ 𝐽 = (LSpan‘𝐶) | |
14 | mapdh.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | 14 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
16 | mapdhc.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
17 | 16 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝐹 ∈ 𝐷) |
18 | mapdh.mn | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
19 | 18 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
20 | mapdhcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
21 | 20 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
22 | mapdh.p | . . 3 ⊢ + = (+g‘𝑈) | |
23 | mapdh.a | . . 3 ⊢ ✚ = (+g‘𝐶) | |
24 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑌 = 0 ) | |
25 | mapdh6k.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝑉) | |
26 | 25 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → 𝑍 ∈ 𝑉) |
27 | mapdh6k.xn | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
28 | 27 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
29 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 22, 23, 24, 26, 28 | mapdh6bN 40514 | . 2 ⊢ ((𝜑 ∧ 𝑌 = 0 ) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
30 | 14 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
31 | 16 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝐹 ∈ 𝐷) |
32 | 18 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
33 | 20 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
34 | mapdh6k.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
35 | 34 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝑌 ∈ 𝑉) |
36 | simpr 486 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → 𝑍 = 0 ) | |
37 | 27 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
38 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 30, 31, 32, 33, 22, 23, 35, 36, 37 | mapdh6cN 40515 | . 2 ⊢ ((𝜑 ∧ 𝑍 = 0 ) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
39 | 14 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
40 | 16 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝐹 ∈ 𝐷) |
41 | 18 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
42 | 20 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
43 | 27 | adantr 482 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
44 | 34 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑌 ∈ 𝑉) |
45 | simprl 770 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑌 ≠ 0 ) | |
46 | eldifsn 4786 | . . . 4 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 )) | |
47 | 44, 45, 46 | sylanbrc 584 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
48 | 25 | adantr 482 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑍 ∈ 𝑉) |
49 | simprr 772 | . . . 4 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑍 ≠ 0 ) | |
50 | eldifsn 4786 | . . . 4 ⊢ (𝑍 ∈ (𝑉 ∖ { 0 }) ↔ (𝑍 ∈ 𝑉 ∧ 𝑍 ≠ 0 )) | |
51 | 48, 49, 50 | sylanbrc 584 | . . 3 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
52 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 39, 40, 41, 42, 22, 23, 43, 47, 51 | mapdh6jN 40522 | . 2 ⊢ ((𝜑 ∧ (𝑌 ≠ 0 ∧ 𝑍 ≠ 0 )) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
53 | 29, 38, 52 | pm2.61da2ne 3031 | 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 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 Vcvv 3475 ∖ cdif 3943 ifcif 4524 {csn 4624 {cpr 4626 〈cotp 4632 ↦ cmpt 5227 ‘cfv 6535 ℩crio 7351 (class class class)co 7396 1st c1st 7960 2nd c2nd 7961 Basecbs 17131 +gcplusg 17184 0gc0g 17372 -gcsg 18808 LSpanclspn 20559 HLchlt 38126 LHypclh 38761 DVecHcdvh 39855 LCDualclcd 40363 mapdcmpd 40401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-riotaBAD 37729 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-ot 4633 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-om 7843 df-1st 7962 df-2nd 7963 df-tpos 8198 df-undef 8245 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-0g 17374 df-mre 17517 df-mrc 17518 df-acs 17520 df-proset 18235 df-poset 18253 df-plt 18270 df-lub 18286 df-glb 18287 df-join 18288 df-meet 18289 df-p0 18365 df-p1 18366 df-lat 18372 df-clat 18439 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-grp 18809 df-minusg 18810 df-sbg 18811 df-subg 18988 df-cntz 19166 df-oppg 19194 df-lsm 19488 df-cmn 19634 df-abl 19635 df-mgp 19971 df-ur 19988 df-ring 20040 df-oppr 20128 df-dvdsr 20149 df-unit 20150 df-invr 20180 df-dvr 20193 df-drng 20295 df-lmod 20450 df-lss 20520 df-lsp 20560 df-lvec 20691 df-lsatoms 37752 df-lshyp 37753 df-lcv 37795 df-lfl 37834 df-lkr 37862 df-ldual 37900 df-oposet 37952 df-ol 37954 df-oml 37955 df-covers 38042 df-ats 38043 df-atl 38074 df-cvlat 38098 df-hlat 38127 df-llines 38275 df-lplanes 38276 df-lvols 38277 df-lines 38278 df-psubsp 38280 df-pmap 38281 df-padd 38573 df-lhyp 38765 df-laut 38766 df-ldil 38881 df-ltrn 38882 df-trl 38936 df-tgrp 39520 df-tendo 39532 df-edring 39534 df-dveca 39780 df-disoa 39806 df-dvech 39856 df-dib 39916 df-dic 39950 df-dih 40006 df-doch 40125 df-djh 40172 df-lcdual 40364 df-mapd 40402 |
This theorem is referenced by: mapdh6N 40524 |
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