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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6jN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 40148. Eliminate (𝑁‘{𝑌}) = (𝑁‘{𝑍}) hypothesis. (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 })) |
Ref | Expression |
---|---|
mapdh6jN | ⊢ (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 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
16 | mapdhc.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
17 | 16 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → 𝐹 ∈ 𝐷) |
18 | mapdh.mn | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
19 | 18 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
20 | mapdhcl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
21 | 20 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
22 | mapdh.p | . . 3 ⊢ + = (+g‘𝑈) | |
23 | mapdh.a | . . 3 ⊢ ✚ = (+g‘𝐶) | |
24 | mapdh6i.xn | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
25 | 24 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
26 | mapdh6i.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
27 | 26 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
28 | mapdh6i.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
29 | 28 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
30 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
31 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 17, 19, 21, 22, 23, 25, 27, 29, 30 | mapdh6iN 40145 | . 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) = (𝑁‘{𝑍})) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
32 | 14 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
33 | 16 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → 𝐹 ∈ 𝐷) |
34 | 18 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) |
35 | 20 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → 𝑋 ∈ (𝑉 ∖ { 0 })) |
36 | 26 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → 𝑌 ∈ (𝑉 ∖ { 0 })) |
37 | 28 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → 𝑍 ∈ (𝑉 ∖ { 0 })) |
38 | 24 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
39 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) | |
40 | eqidd 2738 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝐼‘〈𝑋, 𝐹, 𝑌〉) = (𝐼‘〈𝑋, 𝐹, 𝑌〉)) | |
41 | eqidd 2738 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) | |
42 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 32, 33, 34, 35, 22, 23, 36, 37, 38, 39, 40, 41 | mapdh6aN 40136 | . 2 ⊢ ((𝜑 ∧ (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍})) → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
43 | 31, 42 | pm2.61dane 3030 | 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 2941 Vcvv 3443 ∖ cdif 3905 ifcif 4484 {csn 4584 {cpr 4586 〈cotp 4592 ↦ cmpt 5186 ‘cfv 6493 ℩crio 7306 (class class class)co 7351 1st c1st 7911 2nd c2nd 7912 Basecbs 17043 +gcplusg 17093 0gc0g 17281 -gcsg 18710 LSpanclspn 20385 HLchlt 37750 LHypclh 38385 DVecHcdvh 39479 LCDualclcd 39987 mapdcmpd 40025 |
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 2708 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-riotaBAD 37353 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-of 7609 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-undef 8196 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-struct 16979 df-sets 16996 df-slot 17014 df-ndx 17026 df-base 17044 df-ress 17073 df-plusg 17106 df-mulr 17107 df-sca 17109 df-vsca 17110 df-0g 17283 df-mre 17426 df-mrc 17427 df-acs 17429 df-proset 18144 df-poset 18162 df-plt 18179 df-lub 18195 df-glb 18196 df-join 18197 df-meet 18198 df-p0 18274 df-p1 18275 df-lat 18281 df-clat 18348 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-submnd 18562 df-grp 18711 df-minusg 18712 df-sbg 18713 df-subg 18884 df-cntz 19056 df-oppg 19083 df-lsm 19377 df-cmn 19523 df-abl 19524 df-mgp 19856 df-ur 19873 df-ring 19920 df-oppr 20002 df-dvdsr 20023 df-unit 20024 df-invr 20054 df-dvr 20065 df-drng 20140 df-lmod 20277 df-lss 20346 df-lsp 20386 df-lvec 20517 df-lsatoms 37376 df-lshyp 37377 df-lcv 37419 df-lfl 37458 df-lkr 37486 df-ldual 37524 df-oposet 37576 df-ol 37578 df-oml 37579 df-covers 37666 df-ats 37667 df-atl 37698 df-cvlat 37722 df-hlat 37751 df-llines 37899 df-lplanes 37900 df-lvols 37901 df-lines 37902 df-psubsp 37904 df-pmap 37905 df-padd 38197 df-lhyp 38389 df-laut 38390 df-ldil 38505 df-ltrn 38506 df-trl 38560 df-tgrp 39144 df-tendo 39156 df-edring 39158 df-dveca 39404 df-disoa 39430 df-dvech 39480 df-dib 39540 df-dic 39574 df-dih 39630 df-doch 39749 df-djh 39796 df-lcdual 39988 df-mapd 40026 |
This theorem is referenced by: mapdh6kN 40147 |
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