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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap1l6h | Structured version Visualization version GIF version |
Description: Lemmma for hdmap1l6 40495. Part (6) of [Baer] p. 48 line 2. (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 |
---|---|
hdmap1l6h | ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hdmap1l6.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hdmap1l6.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
3 | hdmap1l6.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
4 | hdmap1l6.p | . . . 4 ⊢ + = (+g‘𝑈) | |
5 | hdmap1l6.s | . . . 4 ⊢ − = (-g‘𝑈) | |
6 | hdmap1l6c.o | . . . 4 ⊢ 0 = (0g‘𝑈) | |
7 | hdmap1l6.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
8 | hdmap1l6.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
9 | hdmap1l6.d | . . . 4 ⊢ 𝐷 = (Base‘𝐶) | |
10 | hdmap1l6.a | . . . 4 ⊢ ✚ = (+g‘𝐶) | |
11 | hdmap1l6.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
12 | hdmap1l6.q | . . . 4 ⊢ 𝑄 = (0g‘𝐶) | |
13 | hdmap1l6.l | . . . 4 ⊢ 𝐿 = (LSpan‘𝐶) | |
14 | hdmap1l6.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
15 | hdmap1l6.i | . . . 4 ⊢ 𝐼 = ((HDMap1‘𝐾)‘𝑊) | |
16 | hdmap1l6.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
17 | hdmap1l6.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
18 | hdmap1l6cl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
19 | hdmap1l6.mn | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐿‘{𝐹})) | |
20 | hdmap1l6d.xn | . . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})) | |
21 | hdmap1l6d.yz | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) = (𝑁‘{𝑍})) | |
22 | hdmap1l6d.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
23 | hdmap1l6d.z | . . . 4 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 })) | |
24 | hdmap1l6d.w | . . . 4 ⊢ (𝜑 → 𝑤 ∈ (𝑉 ∖ { 0 })) | |
25 | hdmap1l6d.wn | . . . 4 ⊢ (𝜑 → ¬ 𝑤 ∈ (𝑁‘{𝑋, 𝑌})) | |
26 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25 | hdmap1l6g 40490 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) = (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑌〉)) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) |
27 | 1, 8, 16 | lcdlmod 40266 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod) |
28 | 1, 2, 16 | dvhlvec 39783 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec) |
29 | 24 | eldifad 3956 | . . . . . . . 8 ⊢ (𝜑 → 𝑤 ∈ 𝑉) |
30 | 18 | eldifad 3956 | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
31 | 22 | eldifad 3956 | . . . . . . . 8 ⊢ (𝜑 → 𝑌 ∈ 𝑉) |
32 | 3, 7, 28, 29, 30, 31, 25 | lspindpi 20694 | . . . . . . 7 ⊢ (𝜑 → ((𝑁‘{𝑤}) ≠ (𝑁‘{𝑋}) ∧ (𝑁‘{𝑤}) ≠ (𝑁‘{𝑌}))) |
33 | 32 | simpld 495 | . . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑤}) ≠ (𝑁‘{𝑋})) |
34 | 33 | necomd 2995 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑤})) |
35 | 1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 34, 18, 29 | hdmap1cl 40478 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑤〉) ∈ 𝐷) |
36 | 23 | eldifad 3956 | . . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉) |
37 | 3, 7, 28, 30, 31, 36, 20 | lspindpi 20694 | . . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))) |
38 | 37 | simpld 495 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
39 | 1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 38, 18, 31 | hdmap1cl 40478 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷) |
40 | 37 | simprd 496 | . . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍})) |
41 | 1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 40, 18, 36 | hdmap1cl 40478 | . . . 4 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) |
42 | 9, 10 | lmodass 20436 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ∈ 𝐷 ∧ (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷 ∧ (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷)) → (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑌〉)) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
43 | 27, 35, 39, 41, 42 | syl13anc 1372 | . . 3 ⊢ (𝜑 → (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑌〉)) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
44 | 26, 43 | eqtrd 2771 | . 2 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
45 | 3, 4, 6, 7, 28, 18, 22, 23, 24, 21, 38, 25 | mapdindp1 40394 | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{(𝑌 + 𝑍)})) |
46 | 1, 2, 16 | dvhlmod 39784 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod) |
47 | 3, 4 | lmodvacl 20435 | . . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉) → (𝑌 + 𝑍) ∈ 𝑉) |
48 | 46, 31, 36, 47 | syl3anc 1371 | . . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) ∈ 𝑉) |
49 | 1, 2, 3, 6, 7, 8, 9, 13, 14, 15, 16, 17, 19, 45, 18, 48 | hdmap1cl 40478 | . . 3 ⊢ (𝜑 → (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) ∈ 𝐷) |
50 | 9, 10 | lmodvacl 20435 | . . . 4 ⊢ ((𝐶 ∈ LMod ∧ (𝐼‘〈𝑋, 𝐹, 𝑌〉) ∈ 𝐷 ∧ (𝐼‘〈𝑋, 𝐹, 𝑍〉) ∈ 𝐷) → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷) |
51 | 27, 39, 41, 50 | syl3anc 1371 | . . 3 ⊢ (𝜑 → ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷) |
52 | 9, 10 | lmodlcan 20437 | . . 3 ⊢ ((𝐶 ∈ LMod ∧ ((𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) ∈ 𝐷 ∧ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)) ∈ 𝐷 ∧ (𝐼‘〈𝑋, 𝐹, 𝑤〉) ∈ 𝐷)) → (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) ↔ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
53 | 27, 49, 51, 35, 52 | syl13anc 1372 | . 2 ⊢ (𝜑 → (((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉)) = ((𝐼‘〈𝑋, 𝐹, 𝑤〉) ✚ ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉))) ↔ (𝐼‘〈𝑋, 𝐹, (𝑌 + 𝑍)〉) = ((𝐼‘〈𝑋, 𝐹, 𝑌〉) ✚ (𝐼‘〈𝑋, 𝐹, 𝑍〉)))) |
54 | 44, 53 | mpbid 231 | 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 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∖ cdif 3941 {csn 4622 {cpr 4624 〈cotp 4630 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 +gcplusg 17179 0gc0g 17367 -gcsg 18796 LModclmod 20420 LSpanclspn 20531 HLchlt 38023 LHypclh 38658 DVecHcdvh 39752 LCDualclcd 40260 mapdcmpd 40298 HDMap1chdma1 40465 |
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 2702 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 ax-cnex 11148 ax-resscn 11149 ax-1cn 11150 ax-icn 11151 ax-addcl 11152 ax-addrcl 11153 ax-mulcl 11154 ax-mulrcl 11155 ax-mulcom 11156 ax-addass 11157 ax-mulass 11158 ax-distr 11159 ax-i2m1 11160 ax-1ne0 11161 ax-1rid 11162 ax-rnegex 11163 ax-rrecex 11164 ax-cnre 11165 ax-pre-lttri 11166 ax-pre-lttrn 11167 ax-pre-ltadd 11168 ax-pre-mulgt0 11169 ax-riotaBAD 37626 |
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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-tp 4627 df-op 4629 df-ot 4631 df-uni 4902 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6289 df-ord 6356 df-on 6357 df-lim 6358 df-suc 6359 df-iota 6484 df-fun 6534 df-fn 6535 df-f 6536 df-f1 6537 df-fo 6538 df-f1o 6539 df-fv 6540 df-riota 7349 df-ov 7396 df-oprab 7397 df-mpo 7398 df-of 7653 df-om 7839 df-1st 7957 df-2nd 7958 df-tpos 8193 df-undef 8240 df-frecs 8248 df-wrecs 8279 df-recs 8353 df-rdg 8392 df-1o 8448 df-er 8686 df-map 8805 df-en 8923 df-dom 8924 df-sdom 8925 df-fin 8926 df-pnf 11232 df-mnf 11233 df-xr 11234 df-ltxr 11235 df-le 11236 df-sub 11428 df-neg 11429 df-nn 12195 df-2 12257 df-3 12258 df-4 12259 df-5 12260 df-6 12261 df-n0 12455 df-z 12541 df-uz 12805 df-fz 13467 df-struct 17062 df-sets 17079 df-slot 17097 df-ndx 17109 df-base 17127 df-ress 17156 df-plusg 17192 df-mulr 17193 df-sca 17195 df-vsca 17196 df-0g 17369 df-mre 17512 df-mrc 17513 df-acs 17515 df-proset 18230 df-poset 18248 df-plt 18265 df-lub 18281 df-glb 18282 df-join 18283 df-meet 18284 df-p0 18360 df-p1 18361 df-lat 18367 df-clat 18434 df-mgm 18543 df-sgrp 18592 df-mnd 18603 df-submnd 18648 df-grp 18797 df-minusg 18798 df-sbg 18799 df-subg 18975 df-cntz 19147 df-oppg 19174 df-lsm 19468 df-cmn 19614 df-abl 19615 df-mgp 19947 df-ur 19964 df-ring 20016 df-oppr 20102 df-dvdsr 20123 df-unit 20124 df-invr 20154 df-dvr 20165 df-drng 20267 df-lmod 20422 df-lss 20492 df-lsp 20532 df-lvec 20663 df-lsatoms 37649 df-lshyp 37650 df-lcv 37692 df-lfl 37731 df-lkr 37759 df-ldual 37797 df-oposet 37849 df-ol 37851 df-oml 37852 df-covers 37939 df-ats 37940 df-atl 37971 df-cvlat 37995 df-hlat 38024 df-llines 38172 df-lplanes 38173 df-lvols 38174 df-lines 38175 df-psubsp 38177 df-pmap 38178 df-padd 38470 df-lhyp 38662 df-laut 38663 df-ldil 38778 df-ltrn 38779 df-trl 38833 df-tgrp 39417 df-tendo 39429 df-edring 39431 df-dveca 39677 df-disoa 39703 df-dvech 39753 df-dib 39813 df-dic 39847 df-dih 39903 df-doch 40022 df-djh 40069 df-lcdual 40261 df-mapd 40299 df-hdmap1 40467 |
This theorem is referenced by: hdmap1l6i 40492 |
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