mapdheq2 - Mathbox for Norm Megill

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Theorem mapdheq2 41712

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 41711	. 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 41093	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ LMod)
24	17	eldifad 3975	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
25	18	eldifad 3975	. . . . . . . . 9 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
26	6, 7, 9, 23, 24, 25	lspsnsub 21023	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) = (𝑁‘{(𝑌 − 𝑋)}))
27	26	fveq2d 6911	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝑀‘(𝑁‘{(𝑌 − 𝑋)})))
28	3, 10, 14	lcdlmod 41575	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ LMod)
29	11, 12, 13, 28, 15, 19	lspsnsub 21023	. . . . . . 7 ⊢ (𝜑 → (𝐽‘{(𝐹𝑅𝐺)}) = (𝐽‘{(𝐺𝑅𝐹)}))
30	27, 29	eqeq12d 2751	. . . . . 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 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 2994	. . . . . 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 41711	. . . 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 1537 ∈ wcel 2106 ≠ wne 2938 Vcvv 3478 ∖ cdif 3960 ifcif 4531 {csn 4631 ⟨cotp 4639 ↦ cmpt 5231 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 1^st c1st 8011 2^nd c2nd 8012 Basecbs 17245 0_gc0g 17486 -_gcsg 18966 LSpanclspn 20987 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 LCDualclcd 41569 mapdcmpd 41607

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570 df-mapd 41608

This theorem is referenced by: mapdheq2biN 41713 mapdh7eN 41731 mapdh7cN 41732 mapdh7fN 41734 mapdh75e 41735

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