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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem27 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 40483. Baer p. 45 line 16: "v(x'-y'') = x'-y'" (with equality swapped). (Contributed by NM, 22-Mar-2015.) |
Ref | Expression |
---|---|
mapdpg.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdpg.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdpg.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdpg.v | ⊢ 𝑉 = (Base‘𝑈) |
mapdpg.s | ⊢ − = (-g‘𝑈) |
mapdpg.z | ⊢ 0 = (0g‘𝑈) |
mapdpg.n | ⊢ 𝑁 = (LSpan‘𝑈) |
mapdpg.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdpg.f | ⊢ 𝐹 = (Base‘𝐶) |
mapdpg.r | ⊢ 𝑅 = (-g‘𝐶) |
mapdpg.j | ⊢ 𝐽 = (LSpan‘𝐶) |
mapdpg.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdpg.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
mapdpg.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
mapdpg.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
mapdpg.ne | ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) |
mapdpg.e | ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) |
mapdpgem25.h1 | ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) |
mapdpgem25.i1 | ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) |
mapdpglem26.a | ⊢ 𝐴 = (Scalar‘𝑈) |
mapdpglem26.b | ⊢ 𝐵 = (Base‘𝐴) |
mapdpglem26.t | ⊢ · = ( ·𝑠 ‘𝐶) |
mapdpglem26.o | ⊢ 𝑂 = (0g‘𝐴) |
Ref | Expression |
---|---|
mapdpglem27 | ⊢ (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpg.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdpg.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
3 | mapdpg.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | mapdpg.v | . . . 4 ⊢ 𝑉 = (Base‘𝑈) | |
5 | mapdpg.s | . . . 4 ⊢ − = (-g‘𝑈) | |
6 | mapdpg.z | . . . 4 ⊢ 0 = (0g‘𝑈) | |
7 | mapdpg.n | . . . 4 ⊢ 𝑁 = (LSpan‘𝑈) | |
8 | mapdpg.c | . . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
9 | mapdpg.f | . . . 4 ⊢ 𝐹 = (Base‘𝐶) | |
10 | mapdpg.r | . . . 4 ⊢ 𝑅 = (-g‘𝐶) | |
11 | mapdpg.j | . . . 4 ⊢ 𝐽 = (LSpan‘𝐶) | |
12 | mapdpg.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
13 | mapdpg.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
14 | mapdpg.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
15 | mapdpg.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
16 | mapdpg.ne | . . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
17 | mapdpg.e | . . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
18 | mapdpgem25.h1 | . . . 4 ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))) | |
19 | mapdpgem25.i1 | . . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) | |
20 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 | mapdpglem25 40474 | . . 3 ⊢ (𝜑 → ((𝐽‘{ℎ}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}))) |
21 | 20 | simprd 497 | . 2 ⊢ (𝜑 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)})) |
22 | eqid 2733 | . . . 4 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
23 | eqid 2733 | . . . 4 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶)) | |
24 | eqid 2733 | . . . 4 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶)) | |
25 | mapdpglem26.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐶) | |
26 | 1, 8, 12 | lcdlvec 40368 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec) |
27 | 1, 8, 12 | lcdlmod 40369 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod) |
28 | 18 | simpld 496 | . . . . 5 ⊢ (𝜑 → ℎ ∈ 𝐹) |
29 | 9, 10 | lmodvsubcl 20494 | . . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ℎ ∈ 𝐹) → (𝐺𝑅ℎ) ∈ 𝐹) |
30 | 27, 15, 28, 29 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → (𝐺𝑅ℎ) ∈ 𝐹) |
31 | 19 | simpld 496 | . . . . 5 ⊢ (𝜑 → 𝑖 ∈ 𝐹) |
32 | 9, 10 | lmodvsubcl 20494 | . . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) → (𝐺𝑅𝑖) ∈ 𝐹) |
33 | 27, 15, 31, 32 | syl3anc 1372 | . . . 4 ⊢ (𝜑 → (𝐺𝑅𝑖) ∈ 𝐹) |
34 | 9, 22, 23, 24, 25, 11, 26, 30, 33 | lspsneq 20712 | . . 3 ⊢ (𝜑 → ((𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}) ↔ ∃𝑣 ∈ ((Base‘(Scalar‘𝐶)) ∖ {(0g‘(Scalar‘𝐶))})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))) |
35 | mapdpglem26.a | . . . . . 6 ⊢ 𝐴 = (Scalar‘𝑈) | |
36 | mapdpglem26.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐴) | |
37 | 1, 3, 35, 36, 8, 22, 23, 12 | lcdsbase 40377 | . . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
38 | mapdpglem26.o | . . . . . . 7 ⊢ 𝑂 = (0g‘𝐴) | |
39 | 1, 3, 35, 38, 8, 22, 24, 12 | lcd0 40385 | . . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = 𝑂) |
40 | 39 | sneqd 4636 | . . . . 5 ⊢ (𝜑 → {(0g‘(Scalar‘𝐶))} = {𝑂}) |
41 | 37, 40 | difeq12d 4121 | . . . 4 ⊢ (𝜑 → ((Base‘(Scalar‘𝐶)) ∖ {(0g‘(Scalar‘𝐶))}) = (𝐵 ∖ {𝑂})) |
42 | 41 | rexeqdv 3327 | . . 3 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘(Scalar‘𝐶)) ∖ {(0g‘(Scalar‘𝐶))})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))) |
43 | 34, 42 | bitrd 279 | . 2 ⊢ (𝜑 → ((𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))) |
44 | 21, 43 | mpbid 231 | 1 ⊢ (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2941 ∃wrex 3071 ∖ cdif 3943 {csn 4624 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 Scalarcsca 17187 ·𝑠 cvsca 17188 0gc0g 17372 -gcsg 18808 LModclmod 20448 LSpanclspn 20559 HLchlt 38126 LHypclh 38761 DVecHcdvh 39855 LCDualclcd 40363 mapdcmpd 40401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pow 5359 ax-pr 5423 ax-un 7712 ax-cnex 11153 ax-resscn 11154 ax-1cn 11155 ax-icn 11156 ax-addcl 11157 ax-addrcl 11158 ax-mulcl 11159 ax-mulrcl 11160 ax-mulcom 11161 ax-addass 11162 ax-mulass 11163 ax-distr 11164 ax-i2m1 11165 ax-1ne0 11166 ax-1rid 11167 ax-rnegex 11168 ax-rrecex 11169 ax-cnre 11170 ax-pre-lttri 11171 ax-pre-lttrn 11172 ax-pre-ltadd 11173 ax-pre-mulgt0 11174 ax-riotaBAD 37729 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4905 df-int 4947 df-iun 4995 df-iin 4996 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-riota 7352 df-ov 7399 df-oprab 7400 df-mpo 7401 df-of 7657 df-om 7843 df-1st 7962 df-2nd 7963 df-tpos 8198 df-undef 8245 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8691 df-map 8810 df-en 8928 df-dom 8929 df-sdom 8930 df-fin 8931 df-pnf 11237 df-mnf 11238 df-xr 11239 df-ltxr 11240 df-le 11241 df-sub 11433 df-neg 11434 df-nn 12200 df-2 12262 df-3 12263 df-4 12264 df-5 12265 df-6 12266 df-n0 12460 df-z 12546 df-uz 12810 df-fz 13472 df-struct 17067 df-sets 17084 df-slot 17102 df-ndx 17114 df-base 17132 df-ress 17161 df-plusg 17197 df-mulr 17198 df-sca 17200 df-vsca 17201 df-0g 17374 df-mre 17517 df-mrc 17518 df-acs 17520 df-proset 18235 df-poset 18253 df-plt 18270 df-lub 18286 df-glb 18287 df-join 18288 df-meet 18289 df-p0 18365 df-p1 18366 df-lat 18372 df-clat 18439 df-mgm 18548 df-sgrp 18597 df-mnd 18613 df-submnd 18659 df-grp 18809 df-minusg 18810 df-sbg 18811 df-subg 18988 df-cntz 19166 df-oppg 19194 df-lsm 19488 df-cmn 19634 df-abl 19635 df-mgp 19971 df-ur 19988 df-ring 20040 df-oppr 20128 df-dvdsr 20149 df-unit 20150 df-invr 20180 df-dvr 20193 df-drng 20295 df-lmod 20450 df-lss 20520 df-lsp 20560 df-lvec 20691 df-lsatoms 37752 df-lshyp 37753 df-lcv 37795 df-lfl 37834 df-lkr 37862 df-ldual 37900 df-oposet 37952 df-ol 37954 df-oml 37955 df-covers 38042 df-ats 38043 df-atl 38074 df-cvlat 38098 df-hlat 38127 df-llines 38275 df-lplanes 38276 df-lvols 38277 df-lines 38278 df-psubsp 38280 df-pmap 38281 df-padd 38573 df-lhyp 38765 df-laut 38766 df-ldil 38881 df-ltrn 38882 df-trl 38936 df-tgrp 39520 df-tendo 39532 df-edring 39534 df-dveca 39780 df-disoa 39806 df-dvech 39856 df-dib 39916 df-dic 39950 df-dih 40006 df-doch 40125 df-djh 40172 df-lcdual 40364 |
This theorem is referenced by: mapdpglem32 40482 |
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