mapdheq2 - Mathbox for Norm Megill

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Theorem mapdheq2 42026

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 42025	. 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 41407	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
24	17	eldifad 3914	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
25	18	eldifad 3914	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
26	6, 7, 9, 23, 24, 25	lspsnsub 20962	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) = (𝑁‘{(𝑌 − 𝑋)}))
27	26	fveq2d 6839	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝑀‘(𝑁‘{(𝑌 − 𝑋)})))
28	3, 10, 14	lcdlmod 41889	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ LMod)
29	11, 12, 13, 28, 15, 19	lspsnsub 20962	. . . . . . 7 ⊢ (𝜑 → (𝐽‘{(𝐹𝑅𝐺)}) = (𝐽‘{(𝐺𝑅𝐹)}))
30	27, 29	eqeq12d 2753	. . . . . 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 717	. . . 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 771	. . . . 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 2988	. . . . . 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 42025	. . . 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 714	. . 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 1542 ∈ wcel 2114 ≠ wne 2933 Vcvv 3441 ∖ cdif 3899 ifcif 4480 {csn 4581 ⟨cotp 4589 ↦ cmpt 5180 ‘cfv 6493 ℩crio 7316 (class class class)co 7360 1^st c1st 7933 2^nd c2nd 7934 Basecbs 17140 0_gc0g 17363 -_gcsg 18869 LSpanclspn 20926 HLchlt 39647 LHypclh 40281 DVecHcdvh 41375 LCDualclcd 41883 mapdcmpd 41921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-riotaBAD 39250

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-ot 4590 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 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 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-of 7624 df-om 7811 df-1st 7935 df-2nd 7936 df-tpos 8170 df-undef 8217 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-n0 12406 df-z 12493 df-uz 12756 df-fz 13428 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17141 df-ress 17162 df-plusg 17194 df-mulr 17195 df-sca 17197 df-vsca 17198 df-0g 17365 df-mre 17509 df-mrc 17510 df-acs 17512 df-proset 18221 df-poset 18240 df-plt 18255 df-lub 18271 df-glb 18272 df-join 18273 df-meet 18274 df-p0 18350 df-p1 18351 df-lat 18359 df-clat 18426 df-mgm 18569 df-sgrp 18648 df-mnd 18664 df-submnd 18713 df-grp 18870 df-minusg 18871 df-sbg 18872 df-subg 19057 df-cntz 19250 df-oppg 19279 df-lsm 19569 df-cmn 19715 df-abl 19716 df-mgp 20080 df-rng 20092 df-ur 20121 df-ring 20174 df-oppr 20277 df-dvdsr 20297 df-unit 20298 df-invr 20328 df-dvr 20341 df-nzr 20450 df-rlreg 20631 df-domn 20632 df-drng 20668 df-lmod 20817 df-lss 20887 df-lsp 20927 df-lvec 21059 df-lsatoms 39273 df-lshyp 39274 df-lcv 39316 df-lfl 39355 df-lkr 39383 df-ldual 39421 df-oposet 39473 df-ol 39475 df-oml 39476 df-covers 39563 df-ats 39564 df-atl 39595 df-cvlat 39619 df-hlat 39648 df-llines 39795 df-lplanes 39796 df-lvols 39797 df-lines 39798 df-psubsp 39800 df-pmap 39801 df-padd 40093 df-lhyp 40285 df-laut 40286 df-ldil 40401 df-ltrn 40402 df-trl 40456 df-tgrp 41040 df-tendo 41052 df-edring 41054 df-dveca 41300 df-disoa 41326 df-dvech 41376 df-dib 41436 df-dic 41470 df-dih 41526 df-doch 41645 df-djh 41692 df-lcdual 41884 df-mapd 41922

This theorem is referenced by: mapdheq2biN 42027 mapdh7eN 42045 mapdh7cN 42046 mapdh7fN 42048 mapdh75e 42049

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