mapdheq4 - Mathbox for Norm Megill

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Theorem mapdheq4 42231

Description: Lemma for ~? mapdh . Part (4) in [Baer] p. 46. (Contributed by NM, 12-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 }))
mapdhe4.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdhe.z	⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
mapdh.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
mapdh.yz	⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
mapdh.eg	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
mapdh.ee	⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)

**Assertion**
Ref	Expression
mapdheq4	⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸)

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 ℎ,𝐸 ℎ,𝑍,𝑥 Allowed substitution hints: 𝜑(𝑥) 𝐶(𝑥) 𝑄(ℎ) 𝑈(𝑥) 𝐸(𝑥) 𝐻(𝑥,ℎ) 𝐼(𝑥,ℎ) 𝐾(𝑥,ℎ) 𝑉(𝑥,ℎ) 𝑊(𝑥,ℎ)

**Proof of Theorem mapdheq4**
Step	Hyp	Ref	Expression
1		mapdh.ee	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸)
2		mapdh.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝐶)
3		mapdh.i	. . . . 5 ⊢ 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)})))))
4		mapdh.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		mapdh.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		mapdh.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		mapdh.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
8		mapdh.s	. . . . 5 ⊢ − = (-_g‘𝑈)
9		mapdhc.o	. . . . 5 ⊢ 0 = (0_g‘𝑈)
10		mapdh.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
11		mapdh.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
12		mapdh.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐶)
13		mapdh.r	. . . . 5 ⊢ 𝑅 = (-_g‘𝐶)
14		mapdh.j	. . . . 5 ⊢ 𝐽 = (LSpan‘𝐶)
15		mapdh.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
16		mapdhc.f	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
17		mapdh.mn	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
18		mapdhcl.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19		mapdhe.z	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ (𝑉 ∖ { 0 }))
20	19	eldifad 3902	. . . . . . 7 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
21	4, 6, 15	dvhlvec 41608	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LVec)
22		mapdhe4.y	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
23	18	eldifad 3902	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
24		mapdh.yz	. . . . . . . . 9 ⊢ (𝜑 → (𝑁‘{𝑌}) ≠ (𝑁‘{𝑍}))
25		mapdh.xn	. . . . . . . . 9 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
26	7, 9, 10, 21, 22, 20, 23, 24, 25	lspindp1 21133	. . . . . . . 8 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}) ∧ ¬ 𝑌 ∈ (𝑁‘{𝑋, 𝑍})))
27	26	simpld 495	. . . . . . 7 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑍}))
28	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 27	mapdhcl 42226	. . . . . 6 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑍⟩) ∈ 𝐷)
29	1, 28	eqeltrrd 2841	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐷)
30	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 29, 27	mapdheq 42227	. . . 4 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑍⟩) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)}))))
31	1, 30	mpbid 233	. . 3 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑍)})) = (𝐽‘{(𝐹𝑅𝐸)})))
32	31	simpld 495	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}))
33		mapdh.eg	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺)
34	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 19, 25, 24, 33, 1	mapdheq4lem 42230	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑌 − 𝑍)})) = (𝐽‘{(𝐺𝑅𝐸)}))
35	22	eldifad 3902	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
36	7, 9, 10, 21, 35, 19, 23, 24, 25	lspindp2 21135	. . . . . 6 ⊢ (𝜑 → ((𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}) ∧ ¬ 𝑍 ∈ (𝑁‘{𝑋, 𝑌})))
37	36	simpld 495	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
38	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 35, 37	mapdhcl 42226	. . . 4 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) ∈ 𝐷)
39	33, 38	eqeltrrd 2841	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
40	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 22, 39, 37	mapdheq 42227	. . . . 5 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))
41	33, 40	mpbid 233	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))
42	41	simpld 495	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
43	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 39, 42, 22, 19, 29, 24	mapdheq 42227	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸 ↔ ((𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}) ∧ (𝑀‘(𝑁‘{(𝑌 − 𝑍)})) = (𝐽‘{(𝐺𝑅𝐸)}))))
44	32, 34, 43	mpbir2and 719	1 ⊢ (𝜑 → (𝐼‘⟨𝑌, 𝐺, 𝑍⟩) = 𝐸)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 Vcvv 3432 ∖ cdif 3887 ifcif 4461 {csn 4562 {cpr 4564 ⟨cotp 4570 ↦ cmpt 5160 ‘cfv 6492 ℩crio 7319 (class class class)co 7363 1^st c1st 7936 2^nd c2nd 7937 Basecbs 17177 0_gc0g 17400 -_gcsg 18909 LSpanclspn 20968 HLchlt 39849 LHypclh 40483 DVecHcdvh 41577 LCDualclcd 42085 mapdcmpd 42123

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-riotaBAD 39452

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-ot 4571 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-tpos 8173 df-undef 8220 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-2o 8403 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17402 df-mre 17546 df-mrc 17547 df-acs 17549 df-proset 18258 df-poset 18277 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18396 df-clat 18463 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-cntz 19290 df-oppg 19319 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-nzr 20492 df-rlreg 20673 df-domn 20674 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lvec 21100 df-lsatoms 39475 df-lshyp 39476 df-lcv 39518 df-lfl 39557 df-lkr 39585 df-ldual 39623 df-oposet 39675 df-ol 39677 df-oml 39678 df-covers 39765 df-ats 39766 df-atl 39797 df-cvlat 39821 df-hlat 39850 df-llines 39997 df-lplanes 39998 df-lvols 39999 df-lines 40000 df-psubsp 40002 df-pmap 40003 df-padd 40295 df-lhyp 40487 df-laut 40488 df-ldil 40603 df-ltrn 40604 df-trl 40658 df-tgrp 41242 df-tendo 41254 df-edring 41256 df-dveca 41502 df-disoa 41528 df-dvech 41578 df-dib 41638 df-dic 41672 df-dih 41728 df-doch 41847 df-djh 41894 df-lcdual 42086 df-mapd 42124

This theorem is referenced by: mapdh7dN 42249 mapdh75d 42253 mapdh8a 42274 hdmap1eq4N 42305

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