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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem27 | Structured version Visualization version GIF version |
Description: Lemma for mapdpg 38857. 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 38848 | . . 3 ⊢ (𝜑 → ((𝐽‘{ℎ}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}))) |
21 | 20 | simprd 498 | . 2 ⊢ (𝜑 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)})) |
22 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶) | |
23 | eqid 2821 | . . . 4 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶)) | |
24 | eqid 2821 | . . . 4 ⊢ (0g‘(Scalar‘𝐶)) = (0g‘(Scalar‘𝐶)) | |
25 | mapdpglem26.t | . . . 4 ⊢ · = ( ·𝑠 ‘𝐶) | |
26 | 1, 8, 12 | lcdlvec 38742 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec) |
27 | 1, 8, 12 | lcdlmod 38743 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod) |
28 | 18 | simpld 497 | . . . . 5 ⊢ (𝜑 → ℎ ∈ 𝐹) |
29 | 9, 10 | lmodvsubcl 19679 | . . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ℎ ∈ 𝐹) → (𝐺𝑅ℎ) ∈ 𝐹) |
30 | 27, 15, 28, 29 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝐺𝑅ℎ) ∈ 𝐹) |
31 | 19 | simpld 497 | . . . . 5 ⊢ (𝜑 → 𝑖 ∈ 𝐹) |
32 | 9, 10 | lmodvsubcl 19679 | . . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) → (𝐺𝑅𝑖) ∈ 𝐹) |
33 | 27, 15, 31, 32 | syl3anc 1367 | . . . 4 ⊢ (𝜑 → (𝐺𝑅𝑖) ∈ 𝐹) |
34 | 9, 22, 23, 24, 25, 11, 26, 30, 33 | lspsneq 19894 | . . 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 38751 | . . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵) |
38 | mapdpglem26.o | . . . . . . 7 ⊢ 𝑂 = (0g‘𝐴) | |
39 | 1, 3, 35, 38, 8, 22, 24, 12 | lcd0 38759 | . . . . . 6 ⊢ (𝜑 → (0g‘(Scalar‘𝐶)) = 𝑂) |
40 | 39 | sneqd 4579 | . . . . 5 ⊢ (𝜑 → {(0g‘(Scalar‘𝐶))} = {𝑂}) |
41 | 37, 40 | difeq12d 4100 | . . . 4 ⊢ (𝜑 → ((Base‘(Scalar‘𝐶)) ∖ {(0g‘(Scalar‘𝐶))}) = (𝐵 ∖ {𝑂})) |
42 | 41 | rexeqdv 3416 | . . 3 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘(Scalar‘𝐶)) ∖ {(0g‘(Scalar‘𝐶))})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))) |
43 | 34, 42 | bitrd 281 | . 2 ⊢ (𝜑 → ((𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))) |
44 | 21, 43 | mpbid 234 | 1 ⊢ (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 ∖ cdif 3933 {csn 4567 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Scalarcsca 16568 ·𝑠 cvsca 16569 0gc0g 16713 -gcsg 18105 LModclmod 19634 LSpanclspn 19743 HLchlt 36501 LHypclh 37135 DVecHcdvh 38229 LCDualclcd 38737 mapdcmpd 38775 |
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 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-0g 16715 df-mre 16857 df-mrc 16858 df-acs 16860 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-oppg 18474 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-lsatoms 36127 df-lshyp 36128 df-lcv 36170 df-lfl 36209 df-lkr 36237 df-ldual 36275 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tgrp 37894 df-tendo 37906 df-edring 37908 df-dveca 38154 df-disoa 38180 df-dvech 38230 df-dib 38290 df-dic 38324 df-dih 38380 df-doch 38499 df-djh 38546 df-lcdual 38738 |
This theorem is referenced by: mapdpglem32 38856 |
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