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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdh6eN | Structured version Visualization version GIF version |
Description: Lemmma for mapdh6N 38439. Part (6) in [Baer] p. 47 line 38. (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) |
mapdh6d.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
mapdh6d.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
mapdh6d.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdh6d.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
mapdh6d.w | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
mapdh6d.wn | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
Ref | Expression |
---|---|
mapdh6eN | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdh.q | . 2 ⊢ 𝑄 = (0g‘𝐶) | |
2 | mapdh.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 | mapdh.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | mapdh.m | . 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
5 | mapdh.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
6 | mapdh.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
7 | mapdh.s | . 2 ⊢ − = (-g‘𝑈) | |
8 | mapdhc.o | . 2 ⊢ 0 = (0g‘𝑈) | |
9 | mapdh.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
10 | mapdh.c | . 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
11 | mapdh.d | . 2 ⊢ 𝐷 = (Base‘𝐶) | |
12 | mapdh.r | . 2 ⊢ 𝑅 = (-g‘𝐶) | |
13 | mapdh.j | . 2 ⊢ 𝐽 = (LSpan‘𝐶) | |
14 | mapdh.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
15 | mapdhc.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
16 | mapdh.mn | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹})) | |
17 | mapdhcl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
18 | mapdh.p | . 2 ⊢ + = (+g‘𝑈) | |
19 | mapdh.a | . 2 ⊢ ✚ = (+g‘𝐶) | |
20 | 3, 5, 14 | dvhlmod 37802 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
21 | mapdh6d.w | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
22 | 21 | eldifad 3875 | . . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
23 | mapdh6d.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
24 | 23 | eldifad 3875 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
25 | 6, 18 | lmodvacl 19343 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) ∈ 𝑉) |
26 | 20, 22, 24, 25 | syl3anc 1364 | . . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ 𝑉) |
27 | 3, 5, 14 | dvhlvec 37801 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
28 | 17 | eldifad 3875 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
29 | mapdh6d.wn | . . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
30 | 6, 9, 27, 22, 28, 24, 29 | lspindpi 19599 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
31 | 30 | simprd 496 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})) |
32 | 6, 18, 8, 9, 20, 22, 24, 31 | lmodindp1 19481 | . . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ≠ 0 ) |
33 | eldifsn 4630 | . . 3 ⊢ ((𝑤 + 𝑌) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑤 + 𝑌) ∈ 𝑉 ∧ (𝑤 + 𝑌) ≠ 0 )) | |
34 | 26, 32, 33 | sylanbrc 583 | . 2 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ (𝑉 ∖ { 0 })) |
35 | mapdh6d.z | . 2 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
36 | 35 | eldifad 3875 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
37 | mapdh6d.yz | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
38 | mapdh6d.xn | . . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
39 | 6, 9, 27, 28, 24, 36, 38 | lspindpi 19599 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
40 | 39 | simpld 495 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
41 | 6, 18, 8, 9, 27, 17, 23, 35, 21, 37, 40, 29 | mapdindp3 38414 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
42 | 6, 18, 8, 9, 27, 17, 23, 35, 21, 37, 40, 29 | mapdindp4 38415 | . . . . 5 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)})) |
43 | 6, 8, 9, 27, 17, 26, 36, 41, 42 | lspindp1 19600 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)}))) |
44 | 43 | simprd 496 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)})) |
45 | prcom 4579 | . . . . 5 ⊢ {(𝑤 + 𝑌), 𝑍} = {𝑍, (𝑤 + 𝑌)} | |
46 | 45 | fveq2i 6546 | . . . 4 ⊢ (𝑁‘{(𝑤 + 𝑌), 𝑍}) = (𝑁‘{𝑍, (𝑤 + 𝑌)}) |
47 | 46 | eleq2i 2874 | . . 3 ⊢ (𝑋 ∈ (𝑁‘{(𝑤 + 𝑌), 𝑍}) ↔ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)})) |
48 | 44, 47 | sylnibr 330 | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{(𝑤 + 𝑌), 𝑍})) |
49 | 6, 9, 27, 36, 28, 26, 42 | lspindpi 19599 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}))) |
50 | 49 | simprd 496 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
51 | 50 | necomd 3039 | . 2 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) ≠ (𝑁‘{𝑍})) |
52 | eqidd 2796 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉) = (𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉)) | |
53 | eqidd 2796 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) = (𝐼‘〈𝑋, 𝐹, 𝑍〉)) | |
54 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 34, 35, 48, 51, 52, 53 | mapdh6aN 38427 | 1 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 Vcvv 3437 ∖ cdif 3860 ifcif 4385 {csn 4476 {cpr 4478 〈cotp 4484 ↦ cmpt 5045 ‘cfv 6230 ℩crio 6981 (class class class)co 7021 1st c1st 7548 2nd c2nd 7549 Basecbs 16317 +gcplusg 16399 0gc0g 16547 -gcsg 17868 LModclmod 19329 LSpanclspn 19438 HLchlt 36042 LHypclh 36676 DVecHcdvh 37770 LCDualclcd 38278 mapdcmpd 38316 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5086 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 ax-riotaBAD 35645 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-ot 4485 df-uni 4750 df-int 4787 df-iun 4831 df-iin 4832 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-of 7272 df-om 7442 df-1st 7550 df-2nd 7551 df-tpos 7748 df-undef 7795 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-1o 7958 df-oadd 7962 df-er 8144 df-map 8263 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-2 11553 df-3 11554 df-4 11555 df-5 11556 df-6 11557 df-n0 11751 df-z 11835 df-uz 12099 df-fz 12748 df-struct 16319 df-ndx 16320 df-slot 16321 df-base 16323 df-sets 16324 df-ress 16325 df-plusg 16412 df-mulr 16413 df-sca 16415 df-vsca 16416 df-0g 16549 df-mre 16691 df-mrc 16692 df-acs 16694 df-proset 17372 df-poset 17390 df-plt 17402 df-lub 17418 df-glb 17419 df-join 17420 df-meet 17421 df-p0 17483 df-p1 17484 df-lat 17490 df-clat 17552 df-mgm 17686 df-sgrp 17728 df-mnd 17739 df-submnd 17780 df-grp 17869 df-minusg 17870 df-sbg 17871 df-subg 18035 df-cntz 18193 df-oppg 18220 df-lsm 18496 df-cmn 18640 df-abl 18641 df-mgp 18935 df-ur 18947 df-ring 18994 df-oppr 19068 df-dvdsr 19086 df-unit 19087 df-invr 19117 df-dvr 19128 df-drng 19199 df-lmod 19331 df-lss 19399 df-lsp 19439 df-lvec 19570 df-lsatoms 35668 df-lshyp 35669 df-lcv 35711 df-lfl 35750 df-lkr 35778 df-ldual 35816 df-oposet 35868 df-ol 35870 df-oml 35871 df-covers 35958 df-ats 35959 df-atl 35990 df-cvlat 36014 df-hlat 36043 df-llines 36190 df-lplanes 36191 df-lvols 36192 df-lines 36193 df-psubsp 36195 df-pmap 36196 df-padd 36488 df-lhyp 36680 df-laut 36681 df-ldil 36796 df-ltrn 36797 df-trl 36851 df-tgrp 37435 df-tendo 37447 df-edring 37449 df-dveca 37695 df-disoa 37721 df-dvech 37771 df-dib 37831 df-dic 37865 df-dih 37921 df-doch 38040 df-djh 38087 df-lcdual 38279 df-mapd 38317 |
This theorem is referenced by: mapdh6gN 38434 |
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