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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6cN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 40200. (Contributed by NM, 24-Apr-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) |
mapdh6c.y | ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
mapdh6c.z | ⊢ (𝜑 → 𝑍 = 0 ) |
mapdh6c.ne | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
Ref | Expression |
---|---|
mapdh6cN | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdh.c | . . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
3 | mapdh.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | lcdlmod 40045 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
5 | lmodgrp 20327 | . . . 4 ⊢ (𝐶 ∈ LMod → 𝐶 ∈ Grp) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐶 ∈ Grp) |
7 | mapdh.q | . . . 4 ⊢ 𝑄 = (0g‘𝐶) | |
8 | 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)}))))) | |
9 | mapdh.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
10 | mapdh.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
11 | mapdh.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
12 | mapdh.s | . . . 4 ⊢ − = (-g‘𝑈) | |
13 | mapdhc.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
14 | mapdh.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
15 | mapdh.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
16 | mapdh.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
17 | mapdh.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
18 | mapdhc.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
19 | mapdh.mn | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
20 | mapdhcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
21 | mapdh6c.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) | |
22 | 1, 10, 3 | dvhlvec 39562 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
23 | 20 | eldifad 3922 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
24 | mapdh6c.z | . . . . . . 7 ⊢ (𝜑 → 𝑍 = 0 ) | |
25 | 1, 10, 3 | dvhlmod 39563 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod) |
26 | 11, 13 | lmod0vcl 20349 | . . . . . . . 8 ⊢ (𝑈 ∈ LMod → 0 ∈ 𝑉) |
27 | 25, 26 | syl 17 | . . . . . . 7 ⊢ (𝜑 → 0 ∈ 𝑉) |
28 | 24, 27 | eqeltrd 2838 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
29 | mapdh6c.ne | . . . . . 6 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
30 | 11, 14, 22, 23, 21, 28, 29 | lspindpi 20591 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
31 | 30 | simpld 495 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
32 | 7, 8, 1, 9, 10, 11, 12, 13, 14, 2, 15, 16, 17, 3, 18, 19, 20, 21, 31 | mapdhcl 40180 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) |
33 | mapdh.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
34 | 15, 33, 7 | grprid 18780 | . . 3 ⊢ ((𝐶 ∈ Grp ∧ (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ 𝑄) = (𝐼‘〈𝑋, 𝐹, 𝑌〉)) |
35 | 6, 32, 34 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ 𝑄) = (𝐼‘〈𝑋, 𝐹, 𝑌〉)) |
36 | 24 | oteq3d 4844 | . . . . 5 ⊢ (𝜑 → 〈𝑋, 𝐹, 𝑍〉 = 〈𝑋, 𝐹, 0 〉) |
37 | 36 | fveq2d 6846 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = (𝐼‘〈𝑋, 𝐹, 0 〉)) |
38 | 7, 8, 13, 20, 18 | mapdhval0 40178 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 0 〉) = 𝑄) |
39 | 37, 38 | eqtrd 2776 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = 𝑄) |
40 | 39 | oveq2d 7372 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ 𝑄)) |
41 | 24 | oveq2d 7372 | . . . . 5 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑌 + 0 )) |
42 | lmodgrp 20327 | . . . . . . 7 ⊢ (𝑈 ∈ LMod → 𝑈 ∈ Grp) | |
43 | 25, 42 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ Grp) |
44 | mapdh.p | . . . . . . 7 ⊢ + = (+g‘𝑈) | |
45 | 11, 44, 13 | grprid 18780 | . . . . . 6 ⊢ ((𝑈 ∈ Grp ∧ 𝑌 ∈ 𝑉) → (𝑌 + 0 ) = 𝑌) |
46 | 43, 21, 45 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (𝑌 + 0 ) = 𝑌) |
47 | 41, 46 | eqtrd 2776 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = 𝑌) |
48 | 47 | oteq3d 4844 | . . 3 ⊢ (𝜑 → 〈𝑋, 𝐹, (𝑌 + 𝑍)〉 = 〈𝑋, 𝐹, 𝑌〉) |
49 | 48 | fveq2d 6846 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = (𝐼‘〈𝑋, 𝐹, 𝑌〉)) |
50 | 35, 40, 49 | 3eqtr4rd 2787 | 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 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 Vcvv 3445 ∖ cdif 3907 ifcif 4486 {csn 4586 {cpr 4588 〈cotp 4594 ↦ cmpt 5188 ‘cfv 6496 ℩crio 7311 (class class class)co 7356 1st c1st 7918 2nd c2nd 7919 Basecbs 17082 +gcplusg 17132 0gc0g 17320 Grpcgrp 18747 -gcsg 18749 LModclmod 20320 LSpanclspn 20430 HLchlt 37802 LHypclh 38437 DVecHcdvh 39531 LCDualclcd 40039 mapdcmpd 40077 |
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 37405 |
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 37428 df-lshyp 37429 df-lcv 37471 df-lfl 37510 df-lkr 37538 df-ldual 37576 df-oposet 37628 df-ol 37630 df-oml 37631 df-covers 37718 df-ats 37719 df-atl 37750 df-cvlat 37774 df-hlat 37803 df-llines 37951 df-lplanes 37952 df-lvols 37953 df-lines 37954 df-psubsp 37956 df-pmap 37957 df-padd 38249 df-lhyp 38441 df-laut 38442 df-ldil 38557 df-ltrn 38558 df-trl 38612 df-tgrp 39196 df-tendo 39208 df-edring 39210 df-dveca 39456 df-disoa 39482 df-dvech 39532 df-dib 39592 df-dic 39626 df-dih 39682 df-doch 39801 df-djh 39848 df-lcdual 40040 df-mapd 40078 |
This theorem is referenced by: mapdh6kN 40199 |
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