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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpg | Structured version Visualization version GIF version |
Description: Part 1 of proof of the first fundamental theorem of projective geometry. Part (1) in [Baer] p. 44. Our notation corresponds to Baer's as follows: 𝑀 for *, 𝑁‘{} for F(), 𝐽‘{} for G(), 𝑋 for x, 𝐺 for x', 𝑌 for y, ℎ for y'. TODO: Rename variables per mapdhval 39732. (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})) |
Ref | Expression |
---|---|
mapdpg | ⊢ (𝜑 → ∃!ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdpg.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdpg.m | . . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
3 | mapdpg.u | . . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | mapdpg.v | . . 3 ⊢ 𝑉 = (Base‘𝑈) | |
5 | mapdpg.s | . . 3 ⊢ − = (-g‘𝑈) | |
6 | mapdpg.z | . . 3 ⊢ 0 = (0g‘𝑈) | |
7 | mapdpg.n | . . 3 ⊢ 𝑁 = (LSpan‘𝑈) | |
8 | mapdpg.c | . . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) | |
9 | mapdpg.f | . . 3 ⊢ 𝐹 = (Base‘𝐶) | |
10 | mapdpg.r | . . 3 ⊢ 𝑅 = (-g‘𝐶) | |
11 | mapdpg.j | . . 3 ⊢ 𝐽 = (LSpan‘𝐶) | |
12 | mapdpg.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
13 | mapdpg.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
14 | mapdpg.y | . . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
15 | mapdpg.g | . . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
16 | mapdpg.ne | . . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) | |
17 | mapdpg.e | . . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺})) | |
18 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | mapdpglem24 39712 | . 2 ⊢ (𝜑 → ∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 | mapdpglem32 39713 | . . . 4 ⊢ ((𝜑 ∧ (ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) ∧ (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) → ℎ = 𝑖) |
20 | 19 | 3exp 1118 | . . 3 ⊢ (𝜑 → ((ℎ ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) → ((((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))) → ℎ = 𝑖))) |
21 | 20 | ralrimivv 3116 | . 2 ⊢ (𝜑 → ∀ℎ ∈ 𝐹 ∀𝑖 ∈ 𝐹 ((((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))) → ℎ = 𝑖)) |
22 | sneq 4577 | . . . . . 6 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖}) | |
23 | 22 | fveq2d 6773 | . . . . 5 ⊢ (ℎ = 𝑖 → (𝐽‘{ℎ}) = (𝐽‘{𝑖})) |
24 | 23 | eqeq2d 2751 | . . . 4 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}))) |
25 | oveq2 7277 | . . . . . . 7 ⊢ (ℎ = 𝑖 → (𝐺𝑅ℎ) = (𝐺𝑅𝑖)) | |
26 | 25 | sneqd 4579 | . . . . . 6 ⊢ (ℎ = 𝑖 → {(𝐺𝑅ℎ)} = {(𝐺𝑅𝑖)}) |
27 | 26 | fveq2d 6773 | . . . . 5 ⊢ (ℎ = 𝑖 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)})) |
28 | 27 | eqeq2d 2751 | . . . 4 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))) |
29 | 24, 28 | anbi12d 631 | . . 3 ⊢ (ℎ = 𝑖 → (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))) |
30 | 29 | reu4 3670 | . 2 ⊢ (∃!ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ↔ (∃ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ∀ℎ ∈ 𝐹 ∀𝑖 ∈ 𝐹 ((((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})) ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))) → ℎ = 𝑖))) |
31 | 18, 21, 30 | sylanbrc 583 | 1 ⊢ (𝜑 → ∃!ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∀wral 3066 ∃wrex 3067 ∃!wreu 3068 ∖ cdif 3889 {csn 4567 ‘cfv 6431 (class class class)co 7269 Basecbs 16908 0gc0g 17146 -gcsg 18575 LSpanclspn 20229 HLchlt 37358 LHypclh 37992 DVecHcdvh 39086 LCDualclcd 39594 mapdcmpd 39632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10926 ax-resscn 10927 ax-1cn 10928 ax-icn 10929 ax-addcl 10930 ax-addrcl 10931 ax-mulcl 10932 ax-mulrcl 10933 ax-mulcom 10934 ax-addass 10935 ax-mulass 10936 ax-distr 10937 ax-i2m1 10938 ax-1ne0 10939 ax-1rid 10940 ax-rnegex 10941 ax-rrecex 10942 ax-cnre 10943 ax-pre-lttri 10944 ax-pre-lttrn 10945 ax-pre-ltadd 10946 ax-pre-mulgt0 10947 ax-riotaBAD 36961 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-of 7525 df-om 7705 df-1st 7822 df-2nd 7823 df-tpos 8031 df-undef 8078 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8479 df-map 8598 df-en 8715 df-dom 8716 df-sdom 8717 df-fin 8718 df-pnf 11010 df-mnf 11011 df-xr 11012 df-ltxr 11013 df-le 11014 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-3 12035 df-4 12036 df-5 12037 df-6 12038 df-n0 12232 df-z 12318 df-uz 12580 df-fz 13237 df-struct 16844 df-sets 16861 df-slot 16879 df-ndx 16891 df-base 16909 df-ress 16938 df-plusg 16971 df-mulr 16972 df-sca 16974 df-vsca 16975 df-0g 17148 df-mre 17291 df-mrc 17292 df-acs 17294 df-proset 18009 df-poset 18027 df-plt 18044 df-lub 18060 df-glb 18061 df-join 18062 df-meet 18063 df-p0 18139 df-p1 18140 df-lat 18146 df-clat 18213 df-mgm 18322 df-sgrp 18371 df-mnd 18382 df-submnd 18427 df-grp 18576 df-minusg 18577 df-sbg 18578 df-subg 18748 df-cntz 18919 df-oppg 18946 df-lsm 19237 df-cmn 19384 df-abl 19385 df-mgp 19717 df-ur 19734 df-ring 19781 df-oppr 19858 df-dvdsr 19879 df-unit 19880 df-invr 19910 df-dvr 19921 df-drng 19989 df-lmod 20121 df-lss 20190 df-lsp 20230 df-lvec 20361 df-lsatoms 36984 df-lshyp 36985 df-lcv 37027 df-lfl 37066 df-lkr 37094 df-ldual 37132 df-oposet 37184 df-ol 37186 df-oml 37187 df-covers 37274 df-ats 37275 df-atl 37306 df-cvlat 37330 df-hlat 37359 df-llines 37506 df-lplanes 37507 df-lvols 37508 df-lines 37509 df-psubsp 37511 df-pmap 37512 df-padd 37804 df-lhyp 37996 df-laut 37997 df-ldil 38112 df-ltrn 38113 df-trl 38167 df-tgrp 38751 df-tendo 38763 df-edring 38765 df-dveca 39011 df-disoa 39037 df-dvech 39087 df-dib 39147 df-dic 39181 df-dih 39237 df-doch 39356 df-djh 39403 df-lcdual 39595 df-mapd 39633 |
This theorem is referenced by: mapdhcl 39735 mapdheq 39736 hdmap1eq 39809 |
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