mapdheq2 - Mathbox for Norm Megill

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Theorem mapdheq2 41838

Description: Lemmma for ~? mapdh . One direction of part (2) in [Baer] p. 45. (Contributed by NM, 4-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 }))
mapdhe.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdhe.g	⊢ (𝜑 → 𝐺 ∈ 𝐷)
mapdh.ne2	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))

**Assertion**
Ref	Expression
mapdheq2	⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹))

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 ℎ,𝐺,𝑥 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 mapdheq2**
Step	Hyp	Ref	Expression
1		mapdh.q	. . 3 ⊢ 𝑄 = (0_g‘𝐶)
2		mapdh.i	. . 3 ⊢ 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)})))))
3		mapdh.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		mapdh.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
5		mapdh.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		mapdh.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
7		mapdh.s	. . 3 ⊢ − = (-_g‘𝑈)
8		mapdhc.o	. . 3 ⊢ 0 = (0_g‘𝑈)
9		mapdh.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
10		mapdh.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
11		mapdh.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
12		mapdh.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
13		mapdh.j	. . 3 ⊢ 𝐽 = (LSpan‘𝐶)
14		mapdh.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		mapdhc.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
16		mapdh.mn	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
17		mapdhcl.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
18		mapdhe.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
19		mapdhe.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
20		mapdh.ne2	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
21	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20	mapdheq 41837	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))
22	16	adantr 480	. . . 4 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
23	3, 5, 14	dvhlmod 41219	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
24	17	eldifad 3911	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
25	18	eldifad 3911	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
26	6, 7, 9, 23, 24, 25	lspsnsub 20950	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) = (𝑁‘{(𝑌 − 𝑋)}))
27	26	fveq2d 6835	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝑀‘(𝑁‘{(𝑌 − 𝑋)})))
28	3, 10, 14	lcdlmod 41701	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ LMod)
29	11, 12, 13, 28, 15, 19	lspsnsub 20950	. . . . . . 7 ⊢ (𝜑 → (𝐽‘{(𝐹𝑅𝐺)}) = (𝐽‘{(𝐺𝑅𝐹)}))
30	27, 29	eqeq12d 2749	. . . . . 6 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}) ↔ (𝑀‘(𝑁‘{(𝑌 − 𝑋)})) = (𝐽‘{(𝐺𝑅𝐹)})))
31	30	biimpa 476	. . . . 5 ⊢ ((𝜑 ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})) → (𝑀‘(𝑁‘{(𝑌 − 𝑋)})) = (𝐽‘{(𝐺𝑅𝐹)}))
32	31	adantrl 716	. . . 4 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → (𝑀‘(𝑁‘{(𝑌 − 𝑋)})) = (𝐽‘{(𝐺𝑅𝐹)}))
33	14	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
34	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → 𝐺 ∈ 𝐷)
35		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
36	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → 𝑌 ∈ (𝑉 ∖ { 0 }))
37	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → 𝑋 ∈ (𝑉 ∖ { 0 }))
38	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → 𝐹 ∈ 𝐷)
39	20	necomd 2985	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑋}))
40	39	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑋}))
41	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 33, 34, 35, 36, 37, 38, 40	mapdheq 41837	. . . 4 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → ((𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹 ↔ ((𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}) ∧ (𝑀‘(𝑁‘{(𝑌 − 𝑋)})) = (𝐽‘{(𝐺𝑅𝐹)}))))
42	22, 32, 41	mpbir2and 713	. . 3 ⊢ ((𝜑 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))) → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹)
43	42	ex 412	. 2 ⊢ (𝜑 → (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})) → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹))
44	21, 43	sylbid 240	1 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 → (𝐼‘⟨𝑌, 𝐺, 𝑋⟩) = 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2930 Vcvv 3438 ∖ cdif 3896 ifcif 4476 {csn 4577 ⟨cotp 4585 ↦ cmpt 5176 ‘cfv 6489 ℩crio 7311 (class class class)co 7355 1^st c1st 7928 2^nd c2nd 7929 Basecbs 17130 0_gc0g 17353 -_gcsg 18858 LSpanclspn 20914 HLchlt 39459 LHypclh 40093 DVecHcdvh 41187 LCDualclcd 41695 mapdcmpd 41733

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11072 ax-resscn 11073 ax-1cn 11074 ax-icn 11075 ax-addcl 11076 ax-addrcl 11077 ax-mulcl 11078 ax-mulrcl 11079 ax-mulcom 11080 ax-addass 11081 ax-mulass 11082 ax-distr 11083 ax-i2m1 11084 ax-1ne0 11085 ax-1rid 11086 ax-rnegex 11087 ax-rrecex 11088 ax-cnre 11089 ax-pre-lttri 11090 ax-pre-lttrn 11091 ax-pre-ltadd 11092 ax-pre-mulgt0 11093 ax-riotaBAD 39062

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-tpos 8165 df-undef 8212 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-en 8879 df-dom 8880 df-sdom 8881 df-fin 8882 df-pnf 11158 df-mnf 11159 df-xr 11160 df-ltxr 11161 df-le 11162 df-sub 11356 df-neg 11357 df-nn 12136 df-2 12198 df-3 12199 df-4 12200 df-5 12201 df-6 12202 df-n0 12392 df-z 12479 df-uz 12743 df-fz 13418 df-struct 17068 df-sets 17085 df-slot 17103 df-ndx 17115 df-base 17131 df-ress 17152 df-plusg 17184 df-mulr 17185 df-sca 17187 df-vsca 17188 df-0g 17355 df-mre 17498 df-mrc 17499 df-acs 17501 df-proset 18210 df-poset 18229 df-plt 18244 df-lub 18260 df-glb 18261 df-join 18262 df-meet 18263 df-p0 18339 df-p1 18340 df-lat 18348 df-clat 18415 df-mgm 18558 df-sgrp 18637 df-mnd 18653 df-submnd 18702 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19046 df-cntz 19239 df-oppg 19268 df-lsm 19558 df-cmn 19704 df-abl 19705 df-mgp 20069 df-rng 20081 df-ur 20110 df-ring 20163 df-oppr 20265 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-nzr 20438 df-rlreg 20619 df-domn 20620 df-drng 20656 df-lmod 20805 df-lss 20875 df-lsp 20915 df-lvec 21047 df-lsatoms 39085 df-lshyp 39086 df-lcv 39128 df-lfl 39167 df-lkr 39195 df-ldual 39233 df-oposet 39285 df-ol 39287 df-oml 39288 df-covers 39375 df-ats 39376 df-atl 39407 df-cvlat 39431 df-hlat 39460 df-llines 39607 df-lplanes 39608 df-lvols 39609 df-lines 39610 df-psubsp 39612 df-pmap 39613 df-padd 39905 df-lhyp 40097 df-laut 40098 df-ldil 40213 df-ltrn 40214 df-trl 40268 df-tgrp 40852 df-tendo 40864 df-edring 40866 df-dveca 41112 df-disoa 41138 df-dvech 41188 df-dib 41248 df-dic 41282 df-dih 41338 df-doch 41457 df-djh 41504 df-lcdual 41696 df-mapd 41734

This theorem is referenced by: mapdheq2biN 41839 mapdh7eN 41857 mapdh7cN 41858 mapdh7fN 41860 mapdh75e 41861

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