mapdhcl - Mathbox for Norm Megill

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Theorem mapdhcl 41330

Description: Lemmma for ~? mapdh . (Contributed by NM, 3-Apr-2015.)

**Hypotheses**
Ref	Expression
mapdh.q	⊢ 𝑄 = (0_g‘𝐶)
mapdh.i	⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
mapdh.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdh.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdh.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdh.v	⊢ 𝑉 = (Base‘𝑈)
mapdh.s	⊢ − = (-_g‘𝑈)
mapdhc.o	⊢ 0 = (0_g‘𝑈)
mapdh.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdh.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdh.d	⊢ 𝐷 = (Base‘𝐶)
mapdh.r	⊢ 𝑅 = (-_g‘𝐶)
mapdh.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdh.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdhc.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdh.mn	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdhcl.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdhc.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mapdh.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))

**Assertion**
Ref	Expression
mapdhcl	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)

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

**Proof of Theorem mapdhcl**
Step	Hyp	Ref	Expression
1		oteq3 4886	. . . 4 ⊢ (𝑌 = 0 → ⟨𝑋, 𝐹, 𝑌⟩ = ⟨𝑋, 𝐹, 0 ⟩)
2	1	fveq2d 6900	. . 3 ⊢ (𝑌 = 0 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (𝐼‘⟨𝑋, 𝐹, 0 ⟩))
3	2	eleq1d 2810	. 2 ⊢ (𝑌 = 0 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷 ↔ (𝐼‘⟨𝑋, 𝐹, 0 ⟩) ∈ 𝐷))
4		mapdh.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
5		mapdh.i	. . . 4 ⊢ 𝐼 = (𝑥 ∈ V ↦ if((2^nd ‘𝑥) = 0 , 𝑄, (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{(2^nd ‘𝑥)})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{((1^st ‘(1^st ‘𝑥)) − (2^nd ‘𝑥))})) = (𝐽‘{((2^nd ‘(1^st ‘𝑥))𝑅ℎ)})))))
6		mapdhcl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
7	6	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → 𝑋 ∈ (𝑉 ∖ { 0 }))
8		mapdhc.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
9	8	adantr 479	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → 𝐹 ∈ 𝐷)
10		mapdhc.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
11	10	anim1i 613	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
12		eldifsn 4792	. . . . 5 ⊢ (𝑌 ∈ (𝑉 ∖ { 0 }) ↔ (𝑌 ∈ 𝑉 ∧ 𝑌 ≠ 0 ))
13	11, 12	sylibr 233	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	4, 5, 7, 9, 13	mapdhval2 41329	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))
15		mapdh.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
16		mapdh.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
17		mapdh.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
18		mapdh.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
19		mapdh.s	. . . . 5 ⊢ − = (-_g‘𝑈)
20		mapdhc.o	. . . . 5 ⊢ 0 = (0_g‘𝑈)
21		mapdh.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
22		mapdh.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
23		mapdh.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐶)
24		mapdh.r	. . . . 5 ⊢ 𝑅 = (-_g‘𝐶)
25		mapdh.j	. . . . 5 ⊢ 𝐽 = (LSpan‘𝐶)
26		mapdh.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
27	26	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
28		mapdh.ne	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
29	28	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
30		mapdh.mn	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
31	30	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
32	15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 7, 13, 9, 29, 31	mapdpg 41309	. . . 4 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → ∃!ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))
33		riotacl 7393	. . . 4 ⊢ (∃!ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})) → (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))) ∈ 𝐷)
34	32, 33	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))) ∈ 𝐷)
35	14, 34	eqeltrd 2825	. 2 ⊢ ((𝜑 ∧ 𝑌 ≠ 0 ) → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
36	4, 5, 20, 6, 8	mapdhval0 41328	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) = 𝑄)
37	15, 22, 23, 4, 26	lcd0vcl 41217	. . 3 ⊢ (𝜑 → 𝑄 ∈ 𝐷)
38	36, 37	eqeltrd 2825	. 2 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 0 ⟩) ∈ 𝐷)
39	3, 35, 38	pm2.61ne 3016	1 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∃!wreu 3361 Vcvv 3461 ∖ cdif 3941 ifcif 4530 {csn 4630 ⟨cotp 4638 ↦ cmpt 5232 ‘cfv 6549 ℩crio 7374 (class class class)co 7419 1^st c1st 7992 2^nd c2nd 7993 Basecbs 17183 0_gc0g 17424 -_gcsg 18900 LSpanclspn 20867 HLchlt 38952 LHypclh 39587 DVecHcdvh 40681 LCDualclcd 41189 mapdcmpd 41227

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-riotaBAD 38555

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-ot 4639 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-sca 17252 df-vsca 17253 df-0g 17426 df-mre 17569 df-mrc 17570 df-acs 17572 df-proset 18290 df-poset 18308 df-plt 18325 df-lub 18341 df-glb 18342 df-join 18343 df-meet 18344 df-p0 18420 df-p1 18421 df-lat 18427 df-clat 18494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-cntz 19280 df-oppg 19309 df-lsm 19603 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-lvec 21000 df-lsatoms 38578 df-lshyp 38579 df-lcv 38621 df-lfl 38660 df-lkr 38688 df-ldual 38726 df-oposet 38778 df-ol 38780 df-oml 38781 df-covers 38868 df-ats 38869 df-atl 38900 df-cvlat 38924 df-hlat 38953 df-llines 39101 df-lplanes 39102 df-lvols 39103 df-lines 39104 df-psubsp 39106 df-pmap 39107 df-padd 39399 df-lhyp 39591 df-laut 39592 df-ldil 39707 df-ltrn 39708 df-trl 39762 df-tgrp 40346 df-tendo 40358 df-edring 40360 df-dveca 40606 df-disoa 40632 df-dvech 40682 df-dib 40742 df-dic 40776 df-dih 40832 df-doch 40951 df-djh 40998 df-lcdual 41190 df-mapd 41228

This theorem is referenced by: mapdheq4lem 41334 mapdheq4 41335 mapdh6lem1N 41336 mapdh6lem2N 41337 mapdh6aN 41338 mapdh6bN 41340 mapdh6cN 41341 mapdh6dN 41342 mapdh6hN 41346 mapdh7eN 41351 mapdh7cN 41352 mapdh7fN 41354 mapdh75e 41355 mapdh75fN 41358 mapdh8aa 41379 mapdh8d0N 41385 mapdh8d 41386 mapdh9a 41392 mapdh9aOLDN 41393 hdmap1cl 41407 hdmap1eulem 41425 hdmap1eulemOLDN 41426

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