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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 41191. (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 41171 | . 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 41172 | . . . 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 1117 | . . 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 3194 | . 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 4634 | . . . . . 6 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖}) | |
23 | 22 | fveq2d 6895 | . . . . 5 ⊢ (ℎ = 𝑖 → (𝐽‘{ℎ}) = (𝐽‘{𝑖})) |
24 | 23 | eqeq2d 2739 | . . . 4 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}))) |
25 | oveq2 7422 | . . . . . . 7 ⊢ (ℎ = 𝑖 → (𝐺𝑅ℎ) = (𝐺𝑅𝑖)) | |
26 | 25 | sneqd 4636 | . . . . . 6 ⊢ (ℎ = 𝑖 → {(𝐺𝑅ℎ)} = {(𝐺𝑅𝑖)}) |
27 | 26 | fveq2d 6895 | . . . . 5 ⊢ (ℎ = 𝑖 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)})) |
28 | 27 | eqeq2d 2739 | . . . 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 3725 | . 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 582 | 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 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∀wral 3057 ∃wrex 3066 ∃!wreu 3370 ∖ cdif 3942 {csn 4624 ‘cfv 6542 (class class class)co 7414 Basecbs 17173 0gc0g 17414 -gcsg 18885 LSpanclspn 20848 HLchlt 38816 LHypclh 39451 DVecHcdvh 40545 LCDualclcd 41053 mapdcmpd 41091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-riotaBAD 38419 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8840 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12497 df-z 12583 df-uz 12847 df-fz 13511 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17416 df-mre 17559 df-mrc 17560 df-acs 17562 df-proset 18280 df-poset 18298 df-plt 18315 df-lub 18331 df-glb 18332 df-join 18333 df-meet 18334 df-p0 18410 df-p1 18411 df-lat 18417 df-clat 18484 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-grp 18886 df-minusg 18887 df-sbg 18888 df-subg 19071 df-cntz 19261 df-oppg 19290 df-lsm 19584 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20266 df-dvdsr 20289 df-unit 20290 df-invr 20320 df-dvr 20333 df-drng 20619 df-lmod 20738 df-lss 20809 df-lsp 20849 df-lvec 20981 df-lsatoms 38442 df-lshyp 38443 df-lcv 38485 df-lfl 38524 df-lkr 38552 df-ldual 38590 df-oposet 38642 df-ol 38644 df-oml 38645 df-covers 38732 df-ats 38733 df-atl 38764 df-cvlat 38788 df-hlat 38817 df-llines 38965 df-lplanes 38966 df-lvols 38967 df-lines 38968 df-psubsp 38970 df-pmap 38971 df-padd 39263 df-lhyp 39455 df-laut 39456 df-ldil 39571 df-ltrn 39572 df-trl 39626 df-tgrp 40210 df-tendo 40222 df-edring 40224 df-dveca 40470 df-disoa 40496 df-dvech 40546 df-dib 40606 df-dic 40640 df-dih 40696 df-doch 40815 df-djh 40862 df-lcdual 41054 df-mapd 41092 |
This theorem is referenced by: mapdhcl 41194 mapdheq 41195 hdmap1eq 41268 |
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