mapdheq - Mathbox for Norm Megill

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Theorem mapdheq 42765

Description: Lemmma for ~? mapdh . The defining equation for h(x,x',y)=y' in part (2) in [Baer] p. 45 line 24. (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
mapdheq	⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))

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

**Proof of Theorem mapdheq**
Step	Hyp	Ref	Expression
1		mapdh.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
2		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)})))))
3		mapdhcl.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
4		mapdhc.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
5		mapdhe.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
6	1, 2, 3, 4, 5	mapdhval2 42763	. . 3 ⊢ (𝜑 → (𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))))
7	6	eqeq1d 2763	. 2 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))) = 𝐺))
8		mapdh.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
9		mapdh.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
10		mapdh.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
11		mapdh.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
12		mapdh.s	. . . 4 ⊢ − = (-_g‘𝑈)
13		mapdhc.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
14		mapdh.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
15		mapdh.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
16		mapdh.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
17		mapdh.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
18		mapdh.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
19		mapdh.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20		mapdh.ne2	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
21		mapdh.mn	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
22	8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 3, 5, 4, 20, 21	mapdpg 42743	. . 3 ⊢ (𝜑 → ∃!ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})))
23		nfv 1947	. . . 4 ⊢ Ⅎℎ𝜑
24		nfcvd 2924	. . . 4 ⊢ (𝜑 → Ⅎℎ𝐺)
25		nfvd 1948	. . . 4 ⊢ (𝜑 → Ⅎℎ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))
26		mapdhe.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
27		sneq 4594	. . . . . . . 8 ⊢ (ℎ = 𝐺 → {ℎ} = {𝐺})
28	27	fveq2d 6887	. . . . . . 7 ⊢ (ℎ = 𝐺 → (𝐽‘{ℎ}) = (𝐽‘{𝐺}))
29	28	eqeq2d 2772	. . . . . 6 ⊢ (ℎ = 𝐺 → ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ↔ (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺})))
30		oveq2 7426	. . . . . . . . 9 ⊢ (ℎ = 𝐺 → (𝐹𝑅ℎ) = (𝐹𝑅𝐺))
31	30	sneqd 4596	. . . . . . . 8 ⊢ (ℎ = 𝐺 → {(𝐹𝑅ℎ)} = {(𝐹𝑅𝐺)})
32	31	fveq2d 6887	. . . . . . 7 ⊢ (ℎ = 𝐺 → (𝐽‘{(𝐹𝑅ℎ)}) = (𝐽‘{(𝐹𝑅𝐺)}))
33	32	eqeq2d 2772	. . . . . 6 ⊢ (ℎ = 𝐺 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}) ↔ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})))
34	29, 33	anbi12d 644	. . . . 5 ⊢ (ℎ = 𝐺 → (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))
35	34	adantl 487	. . . 4 ⊢ ((𝜑 ∧ ℎ = 𝐺) → (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)})) ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))
36	23, 24, 25, 26, 35	riota2df 7398	. . 3 ⊢ ((𝜑 ∧ ∃!ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))) → (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})) ↔ (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))) = 𝐺))
37	22, 36	mpdan 700	. 2 ⊢ (𝜑 → (((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)})) ↔ (℩ℎ ∈ 𝐷 ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅ℎ)}))) = 𝐺))
38	7, 37	bitr4d 285	1 ⊢ (𝜑 → ((𝐼‘⟨𝑋, 𝐹, 𝑌⟩) = 𝐺 ↔ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐹𝑅𝐺)}))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2956 ∃!wreu 3364 Vcvv 3451 ∖ cdif 3896 ifcif 4482 {csn 4584 ⟨cotp 4592 ↦ cmpt 5186 ‘cfv 6537 ℩crio 7374 (class class class)co 7418 1^st c1st 7997 2^nd c2nd 7998 Basecbs 17380 0_gc0g 17603 -_gcsg 19139 LSpanclspn 21239 HLchlt 40387 LHypclh 41021 DVecHcdvh 42115 LCDualclcd 42623 mapdcmpd 42661

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-riotaBAD 39990

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-tpos 8236 df-undef 8283 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-n0 12600 df-z 12687 df-uz 12959 df-fz 13633 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-sca 17437 df-vsca 17438 df-0g 17605 df-mre 17749 df-mrc 17750 df-acs 17752 df-proset 18461 df-poset 18480 df-plt 18495 df-lub 18511 df-glb 18512 df-join 18513 df-meet 18514 df-p0 18590 df-p1 18591 df-lat 18599 df-clat 18666 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-grp 19140 df-minusg 19141 df-sbg 19142 df-subg 19326 df-cntz 19524 df-oppg 19553 df-lsm 19843 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-oppr 20560 df-dvdsr 20580 df-unit 20581 df-invr 20611 df-dvr 20624 df-nzr 20756 df-rlreg 20939 df-domn 20940 df-drng 20975 df-lmod 21130 df-lss 21200 df-lsp 21240 df-lvec 21371 df-lsatoms 40013 df-lshyp 40014 df-lcv 40056 df-lfl 40095 df-lkr 40123 df-ldual 40161 df-oposet 40213 df-ol 40215 df-oml 40216 df-covers 40303 df-ats 40304 df-atl 40335 df-cvlat 40359 df-hlat 40388 df-llines 40535 df-lplanes 40536 df-lvols 40537 df-lines 40538 df-psubsp 40540 df-pmap 40541 df-padd 40833 df-lhyp 41025 df-laut 41026 df-ldil 41141 df-ltrn 41142 df-trl 41196 df-tgrp 41780 df-tendo 41792 df-edring 41794 df-dveca 42040 df-disoa 42066 df-dvech 42116 df-dib 42176 df-dic 42210 df-dih 42266 df-doch 42385 df-djh 42432 df-lcdual 42624 df-mapd 42662

This theorem is used by: mapdheq2 42766 mapdheq4lem 42768 mapdheq4 42769 mapdh6lem1N 42770 mapdh6lem2N 42771 mapdh6aN 42772 mapdh7fN 42788 mapdh75fN 42792 mapdh8aa 42813 mapdh8d0N 42819 mapdh8d 42820 mapdh9a 42826 mapdh9aOLDN 42827

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