mapdcnvid2 - Mathbox for Norm Megill

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Theorem mapdcnvid2 42431

Description: Value of the converse of the map defined by df-mapd 42399. (Contributed by NM, 13-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdcnvid2.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdcnvid2.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdcnvid2.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdcnvid2.x	⊢ (𝜑 → 𝑋 ∈ ran 𝑀)

**Assertion**
Ref	Expression
mapdcnvid2	⊢ (𝜑 → (𝑀‘(^◡𝑀‘𝑋)) = 𝑋)

Proof of Theorem mapdcnvid2 Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapdcnvid2.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2763	. . . 4 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
3		mapdcnvid2.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
4		eqid 2763	. . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
5		eqid 2763	. . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
6		eqid 2763	. . . 4 ⊢ (LFnl‘((DVecH‘𝐾)‘𝑊)) = (LFnl‘((DVecH‘𝐾)‘𝑊))
7		eqid 2763	. . . 4 ⊢ (LKer‘((DVecH‘𝐾)‘𝑊)) = (LKer‘((DVecH‘𝐾)‘𝑊))
8		eqid 2763	. . . 4 ⊢ (LDual‘((DVecH‘𝐾)‘𝑊)) = (LDual‘((DVecH‘𝐾)‘𝑊))
9		eqid 2763	. . . 4 ⊢ (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))) = (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊)))
10		eqid 2763	. . . 4 ⊢ {𝑔 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔)} = {𝑔 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔)}
11		mapdcnvid2.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	mapd1o 42422	. . 3 ⊢ (𝜑 → 𝑀:(LSubSp‘((DVecH‘𝐾)‘𝑊))–1-1-onto→((LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))) ∩ 𝒫 {𝑔 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔)}))
13		f1of1 6819	. . 3 ⊢ (𝑀:(LSubSp‘((DVecH‘𝐾)‘𝑊))–1-1-onto→((LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))) ∩ 𝒫 {𝑔 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔)}) → 𝑀:(LSubSp‘((DVecH‘𝐾)‘𝑊))–1-1→((LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))) ∩ 𝒫 {𝑔 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔)}))
14		f1f1orn 6832	. . 3 ⊢ (𝑀:(LSubSp‘((DVecH‘𝐾)‘𝑊))–1-1→((LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))) ∩ 𝒫 {𝑔 ∈ (LFnl‘((DVecH‘𝐾)‘𝑊)) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔))) = ((LKer‘((DVecH‘𝐾)‘𝑊))‘𝑔)}) → 𝑀:(LSubSp‘((DVecH‘𝐾)‘𝑊))–1-1-onto→ran 𝑀)
15	12, 13, 14	3syl 19	. 2 ⊢ (𝜑 → 𝑀:(LSubSp‘((DVecH‘𝐾)‘𝑊))–1-1-onto→ran 𝑀)
16		mapdcnvid2.x	. 2 ⊢ (𝜑 → 𝑋 ∈ ran 𝑀)
17		f1ocnvfv2 7275	. 2 ⊢ ((𝑀:(LSubSp‘((DVecH‘𝐾)‘𝑊))–1-1-onto→ran 𝑀 ∧ 𝑋 ∈ ran 𝑀) → (𝑀‘(^◡𝑀‘𝑋)) = 𝑋)
18	15, 16, 17	syl2anc 595	1 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 {crab 3416 ∩ cin 3904 𝒫 cpw 4562 ^◡ccnv 5660 ran crn 5662 –1-1→wf1 6533 –1-1-onto→wf1o 6535 ‘cfv 6536 LSubSpclss 21052 LFnlclfn 39831 LKerclk 39859 LDualcld 39897 HLchlt 40124 LHypclh 40758 DVecHcdvh 41852 ocHcoch 42121 mapdcmpd 42398

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-riotaBAD 39727

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-undef 8265 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-n0 12500 df-z 12587 df-uz 12858 df-fz 13531 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-sca 17321 df-vsca 17322 df-0g 17489 df-mre 17633 df-mrc 17634 df-acs 17636 df-proset 18345 df-poset 18364 df-plt 18379 df-lub 18395 df-glb 18396 df-join 18397 df-meet 18398 df-p0 18474 df-p1 18475 df-lat 18483 df-clat 18550 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-grp 18998 df-minusg 18999 df-sbg 19000 df-subg 19184 df-cntz 19382 df-oppg 19411 df-lsm 19701 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-nzr 20610 df-rlreg 20793 df-domn 20794 df-drng 20829 df-lmod 20983 df-lss 21053 df-lsp 21093 df-lvec 21224 df-lsatoms 39750 df-lshyp 39751 df-lcv 39793 df-lfl 39832 df-lkr 39860 df-ldual 39898 df-oposet 39950 df-ol 39952 df-oml 39953 df-covers 40040 df-ats 40041 df-atl 40072 df-cvlat 40096 df-hlat 40125 df-llines 40272 df-lplanes 40273 df-lvols 40274 df-lines 40275 df-psubsp 40277 df-pmap 40278 df-padd 40570 df-lhyp 40762 df-laut 40763 df-ldil 40878 df-ltrn 40879 df-trl 40933 df-tgrp 41517 df-tendo 41529 df-edring 41531 df-dveca 41777 df-disoa 41803 df-dvech 41853 df-dib 41913 df-dic 41947 df-dih 42003 df-doch 42122 df-djh 42169 df-mapd 42399

This theorem is referenced by: mapdcnvordN 42432 mapdcv 42434 mapdin 42436 mapdlsm 42438 mapdcnvatN 42440 hdmaprnlem3N 42624 hdmaprnlem9N 42631 hdmaprnlem16N 42636

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