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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1l6e | Structured version Visualization version GIF version |
Description: Lemmma for hdmap1l6 38959. Part (6) in [Baer] p. 47 line 38. (Contributed by NM, 1-May-2015.) |
Ref | Expression |
---|---|
hdmap1l6.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hdmap1l6.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
hdmap1l6.v | ⊢ 𝑉 = (Base‘𝑈) |
hdmap1l6.p | ⊢ + = (+g‘𝑈) |
hdmap1l6.s | ⊢ − = (-g‘𝑈) |
hdmap1l6c.o | ⊢ 0 = (0g‘𝑈) |
hdmap1l6.n | ⊢ 𝑁 = (LSpan‘𝑈) |
hdmap1l6.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
hdmap1l6.d | ⊢ 𝐷 = (Base‘𝐶) |
hdmap1l6.a | ⊢ ✚ = (+g‘𝐶) |
hdmap1l6.r | ⊢ 𝑅 = (-g‘𝐶) |
hdmap1l6.q | ⊢ 𝑄 = (0g‘𝐶) |
hdmap1l6.l | ⊢ 𝐿 = (LSpan‘𝐶) |
hdmap1l6.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
hdmap1l6.i | ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) |
hdmap1l6.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hdmap1l6.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
hdmap1l6cl.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6.mn | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) |
hdmap1l6d.xn | ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) |
hdmap1l6d.yz | ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) |
hdmap1l6d.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6d.z | ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6d.w | ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) |
hdmap1l6d.wn | ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) |
Ref | Expression |
---|---|
hdmap1l6e | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, ((𝑤 + 𝑌) + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap1l6.h | . 2 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap1l6.u | . 2 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap1l6.v | . 2 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap1l6.p | . 2 ⊢ + = (+g‘𝑈) | |
5 | hdmap1l6.s | . 2 ⊢ − = (-g‘𝑈) | |
6 | hdmap1l6c.o | . 2 ⊢ 0 = (0g‘𝑈) | |
7 | hdmap1l6.n | . 2 ⊢ 𝑁 = (LSpan‘𝑈) | |
8 | hdmap1l6.c | . 2 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
9 | hdmap1l6.d | . 2 ⊢ 𝐷 = (Base‘𝐶) | |
10 | hdmap1l6.a | . 2 ⊢ ✚ = (+g‘𝐶) | |
11 | hdmap1l6.r | . 2 ⊢ 𝑅 = (-g‘𝐶) | |
12 | hdmap1l6.q | . 2 ⊢ 𝑄 = (0g‘𝐶) | |
13 | hdmap1l6.l | . 2 ⊢ 𝐿 = (LSpan‘𝐶) | |
14 | hdmap1l6.m | . 2 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
15 | hdmap1l6.i | . 2 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
16 | hdmap1l6.k | . 2 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | hdmap1l6.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
18 | hdmap1l6cl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
19 | hdmap1l6.mn | . 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) | |
20 | 1, 2, 16 | dvhlmod 38248 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod) |
21 | hdmap1l6d.w | . . . . 5 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
22 | 21 | eldifad 3950 | . . . 4 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
23 | hdmap1l6d.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
24 | 23 | eldifad 3950 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
25 | 3, 4 | lmodvacl 19650 | . . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑤 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑤 + 𝑌) ∈ 𝑉) |
26 | 20, 22, 24, 25 | syl3anc 1367 | . . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ 𝑉) |
27 | 1, 2, 16 | dvhlvec 38247 | . . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LVec) |
28 | 18 | eldifad 3950 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
29 | hdmap1l6d.wn | . . . . . 6 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
30 | 3, 7, 27, 22, 28, 24, 29 | lspindpi 19906 | . . . . 5 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
31 | 30 | simprd 498 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌})) |
32 | 3, 4, 6, 7, 20, 22, 24, 31 | lmodindp1 19788 | . . 3 ⊢ (𝜑 → (𝑤 + 𝑌) ≠ 0 ) |
33 | eldifsn 4721 | . . 3 ⊢ ((𝑤 + 𝑌) ∈ (𝑉 ∖ { 0 }) ↔ ((𝑤 + 𝑌) ∈ 𝑉 ∧ (𝑤 + 𝑌) ≠ 0 )) | |
34 | 26, 32, 33 | sylanbrc 585 | . 2 ⊢ (𝜑 → (𝑤 + 𝑌) ∈ (𝑉 ∖ { 0 })) |
35 | hdmap1l6d.z | . 2 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
36 | 35 | eldifad 3950 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
37 | hdmap1l6d.yz | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
38 | hdmap1l6d.xn | . . . . . . . 8 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
39 | 3, 7, 27, 28, 24, 36, 38 | lspindpi 19906 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
40 | 39 | simpld 497 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
41 | 3, 4, 6, 7, 27, 18, 23, 35, 21, 37, 40, 29 | mapdindp3 38860 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
42 | 3, 4, 6, 7, 27, 18, 23, 35, 21, 37, 40, 29 | mapdindp4 38861 | . . . . 5 ⊢ (𝜑 → ¬ 𝑍 ∈ (𝑁‘{𝑋, (𝑤 + 𝑌)})) |
43 | 3, 6, 7, 27, 18, 26, 36, 41, 42 | lspindp1 19907 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}) ∧ ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)}))) |
44 | 43 | simprd 498 | . . 3 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)})) |
45 | prcom 4670 | . . . . 5 ⊢ {(𝑤 + 𝑌), 𝑍} = {𝑍, (𝑤 + 𝑌)} | |
46 | 45 | fveq2i 6675 | . . . 4 ⊢ (𝑁‘{(𝑤 + 𝑌), 𝑍}) = (𝑁‘{𝑍, (𝑤 + 𝑌)}) |
47 | 46 | eleq2i 2906 | . . 3 ⊢ (𝑋 ∈ (𝑁‘{(𝑤 + 𝑌), 𝑍}) ↔ 𝑋 ∈ (𝑁‘{𝑍, (𝑤 + 𝑌)})) |
48 | 44, 47 | sylnibr 331 | . 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{(𝑤 + 𝑌), 𝑍})) |
49 | 3, 7, 27, 36, 28, 26, 42 | lspindpi 19906 | . . . 4 ⊢ (𝜑 → ((𝑁‘{𝑍}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)}))) |
50 | 49 | simprd 498 | . . 3 ⊢ (𝜑 → (𝑁‘{𝑍}) ≠ (𝑁‘{(𝑤 + 𝑌)})) |
51 | 50 | necomd 3073 | . 2 ⊢ (𝜑 → (𝑁‘{(𝑤 + 𝑌)}) ≠ (𝑁‘{𝑍})) |
52 | eqidd 2824 | . 2 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉) = (𝐼‘〈𝑋, 𝐹, (𝑤 + 𝑌)〉)) | |
53 | eqidd 2824 | . 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 | hdmap1l6a 38947 | 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 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 {csn 4569 {cpr 4571 〈cotp 4577 ‘cfv 6357 (class class class)co 7158 Basecbs 16485 +gcplusg 16567 0gc0g 16715 -gcsg 18107 LModclmod 19636 LSpanclspn 19745 HLchlt 36488 LHypclh 37122 DVecHcdvh 38216 LCDualclcd 38724 mapdcmpd 38762 HDMap1chdma1 38929 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-riotaBAD 36091 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-ot 4578 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-undef 7941 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-sca 16583 df-vsca 16584 df-0g 16717 df-mre 16859 df-mrc 16860 df-acs 16862 df-proset 17540 df-poset 17558 df-plt 17570 df-lub 17586 df-glb 17587 df-join 17588 df-meet 17589 df-p0 17651 df-p1 17652 df-lat 17658 df-clat 17720 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-sbg 18110 df-subg 18278 df-cntz 18449 df-oppg 18476 df-lsm 18763 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-lmod 19638 df-lss 19706 df-lsp 19746 df-lvec 19877 df-lsatoms 36114 df-lshyp 36115 df-lcv 36157 df-lfl 36196 df-lkr 36224 df-ldual 36262 df-oposet 36314 df-ol 36316 df-oml 36317 df-covers 36404 df-ats 36405 df-atl 36436 df-cvlat 36460 df-hlat 36489 df-llines 36636 df-lplanes 36637 df-lvols 36638 df-lines 36639 df-psubsp 36641 df-pmap 36642 df-padd 36934 df-lhyp 37126 df-laut 37127 df-ldil 37242 df-ltrn 37243 df-trl 37297 df-tgrp 37881 df-tendo 37893 df-edring 37895 df-dveca 38141 df-disoa 38167 df-dvech 38217 df-dib 38277 df-dic 38311 df-dih 38367 df-doch 38486 df-djh 38533 df-lcdual 38725 df-mapd 38763 df-hdmap1 38931 |
This theorem is referenced by: hdmap1l6g 38954 |
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