mapdpg - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpg		Structured version Visualization version GIF version

Theorem mapdpg 42731

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 42749. (Contributed by NM, 22-Mar-2015.)

**Hypotheses**
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 = (0_g‘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}))

**Assertion**
Ref	Expression
mapdpg	⊢ (𝜑 → ∃!ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))

Distinct variable groups: 𝐶,ℎ ℎ,𝐹 ℎ,𝐺 ℎ,𝐽 ℎ,𝑀 ℎ,𝑁 𝑅,ℎ − ,ℎ 𝑈,ℎ ℎ,𝑋 ℎ,𝑌 𝜑,ℎ Allowed substitution hints: 𝐻(ℎ) 𝐾(ℎ) 𝑉(ℎ) 𝑊(ℎ) 0 (ℎ)

Proof of Theorem mapdpg Dummy variable 𝑖 is distinct from all other variables.

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 = (0_g‘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 42729	. 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 42730	. . . 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 1137	. . 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 3204	. 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 4594	. . . . . 6 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖})
23	22	fveq2d 6881	. . . . 5 ⊢ (ℎ = 𝑖 → (𝐽‘{ℎ}) = (𝐽‘{𝑖}))
24	23	eqeq2d 2772	. . . 4 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖})))
25		oveq2 7420	. . . . . . 7 ⊢ (ℎ = 𝑖 → (𝐺𝑅ℎ) = (𝐺𝑅𝑖))
26	25	sneqd 4596	. . . . . 6 ⊢ (ℎ = 𝑖 → {(𝐺𝑅ℎ)} = {(𝐺𝑅𝑖)})
27	26	fveq2d 6881	. . . . 5 ⊢ (ℎ = 𝑖 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}))
28	27	eqeq2d 2772	. . . 4 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))
29	24, 28	anbi12d 644	. . 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 3689	. 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 595	1 ⊢ (𝜑 → ∃!ℎ ∈ 𝐹 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)})))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 ∃!wreu 3364 ∖ cdif 3896 {csn 4584 ‘cfv 6531 (class class class)co 7412 Basecbs 17367 0_gc0g 17590 -_gcsg 19126 LSpanclspn 21226 HLchlt 40375 LHypclh 41009 DVecHcdvh 42103 LCDualclcd 42611 mapdcmpd 42649

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258 ax-riotaBAD 39978

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7682 df-om 7867 df-1st 7990 df-2nd 7991 df-tpos 8227 df-undef 8274 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8460 df-2o 8461 df-er 8701 df-map 8833 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-nn 12317 df-2 12386 df-3 12387 df-4 12388 df-5 12389 df-6 12390 df-n0 12588 df-z 12675 df-uz 12947 df-fz 13621 df-struct 17305 df-sets 17322 df-slot 17340 df-ndx 17352 df-base 17368 df-ress 17389 df-plusg 17421 df-mulr 17422 df-sca 17424 df-vsca 17425 df-0g 17592 df-mre 17736 df-mrc 17737 df-acs 17739 df-proset 18448 df-poset 18467 df-plt 18482 df-lub 18498 df-glb 18499 df-join 18500 df-meet 18501 df-p0 18577 df-p1 18578 df-lat 18586 df-clat 18653 df-mgm 18796 df-sgrp 18888 df-mnd 18904 df-submnd 18959 df-grp 19127 df-minusg 19128 df-sbg 19129 df-subg 19313 df-cntz 19511 df-oppg 19540 df-lsm 19830 df-cmn 19976 df-abl 19977 df-mgp 20341 df-rng 20355 df-ur 20388 df-ring 20441 df-oppr 20547 df-dvdsr 20567 df-unit 20568 df-invr 20598 df-dvr 20611 df-nzr 20743 df-rlreg 20926 df-domn 20927 df-drng 20962 df-lmod 21117 df-lss 21187 df-lsp 21227 df-lvec 21358 df-lsatoms 40001 df-lshyp 40002 df-lcv 40044 df-lfl 40083 df-lkr 40111 df-ldual 40149 df-oposet 40201 df-ol 40203 df-oml 40204 df-covers 40291 df-ats 40292 df-atl 40323 df-cvlat 40347 df-hlat 40376 df-llines 40523 df-lplanes 40524 df-lvols 40525 df-lines 40526 df-psubsp 40528 df-pmap 40529 df-padd 40821 df-lhyp 41013 df-laut 41014 df-ldil 41129 df-ltrn 41130 df-trl 41184 df-tgrp 41768 df-tendo 41780 df-edring 41782 df-dveca 42028 df-disoa 42054 df-dvech 42104 df-dib 42164 df-dic 42198 df-dih 42254 df-doch 42373 df-djh 42420 df-lcdual 42612 df-mapd 42650

This theorem is used by: mapdhcl 42752 mapdheq 42753 hdmap1eq 42826

W3C validator