mapdpg - Mathbox for Norm Megill

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Theorem mapdpg 41683

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 41701. (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 41681	. 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 41682	. . . 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 1119	. . 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 3187	. 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 4616	. . . . . 6 ⊢ (ℎ = 𝑖 → {ℎ} = {𝑖})
23	22	fveq2d 6890	. . . . 5 ⊢ (ℎ = 𝑖 → (𝐽‘{ℎ}) = (𝐽‘{𝑖}))
24	23	eqeq2d 2745	. . . 4 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖})))
25		oveq2 7421	. . . . . . 7 ⊢ (ℎ = 𝑖 → (𝐺𝑅ℎ) = (𝐺𝑅𝑖))
26	25	sneqd 4618	. . . . . 6 ⊢ (ℎ = 𝑖 → {(𝐺𝑅ℎ)} = {(𝐺𝑅𝑖)})
27	26	fveq2d 6890	. . . . 5 ⊢ (ℎ = 𝑖 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}))
28	27	eqeq2d 2745	. . . 4 ⊢ (ℎ = 𝑖 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)})))
29	24, 28	anbi12d 632	. . 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 3719	. 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 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ∀wral 3050 ∃wrex 3059 ∃!wreu 3361 ∖ cdif 3928 {csn 4606 ‘cfv 6541 (class class class)co 7413 Basecbs 17230 0_gc0g 17456 -_gcsg 18923 LSpanclspn 20938 HLchlt 39326 LHypclh 39961 DVecHcdvh 41055 LCDualclcd 41563 mapdcmpd 41601

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-riotaBAD 38929

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7870 df-1st 7996 df-2nd 7997 df-tpos 8233 df-undef 8280 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8727 df-map 8850 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-n0 12510 df-z 12597 df-uz 12861 df-fz 13530 df-struct 17167 df-sets 17184 df-slot 17202 df-ndx 17214 df-base 17231 df-ress 17254 df-plusg 17287 df-mulr 17288 df-sca 17290 df-vsca 17291 df-0g 17458 df-mre 17601 df-mrc 17602 df-acs 17604 df-proset 18311 df-poset 18330 df-plt 18345 df-lub 18361 df-glb 18362 df-join 18363 df-meet 18364 df-p0 18440 df-p1 18441 df-lat 18447 df-clat 18514 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-subg 19111 df-cntz 19305 df-oppg 19334 df-lsm 19623 df-cmn 19769 df-abl 19770 df-mgp 20107 df-rng 20119 df-ur 20148 df-ring 20201 df-oppr 20303 df-dvdsr 20326 df-unit 20327 df-invr 20357 df-dvr 20370 df-nzr 20482 df-rlreg 20663 df-domn 20664 df-drng 20700 df-lmod 20829 df-lss 20899 df-lsp 20939 df-lvec 21071 df-lsatoms 38952 df-lshyp 38953 df-lcv 38995 df-lfl 39034 df-lkr 39062 df-ldual 39100 df-oposet 39152 df-ol 39154 df-oml 39155 df-covers 39242 df-ats 39243 df-atl 39274 df-cvlat 39298 df-hlat 39327 df-llines 39475 df-lplanes 39476 df-lvols 39477 df-lines 39478 df-psubsp 39480 df-pmap 39481 df-padd 39773 df-lhyp 39965 df-laut 39966 df-ldil 40081 df-ltrn 40082 df-trl 40136 df-tgrp 40720 df-tendo 40732 df-edring 40734 df-dveca 40980 df-disoa 41006 df-dvech 41056 df-dib 41116 df-dic 41150 df-dih 41206 df-doch 41325 df-djh 41372 df-lcdual 41564 df-mapd 41602

This theorem is referenced by: mapdhcl 41704 mapdheq 41705 hdmap1eq 41778

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