mapdcnvid2 - Mathbox for Norm Megill

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Theorem mapdcnvid2 41856

Description: Value of the converse of the map defined by df-mapd 41824. (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 2734	. . . 4 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
3		mapdcnvid2.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
4		eqid 2734	. . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
5		eqid 2734	. . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
6		eqid 2734	. . . 4 ⊢ (LFnl‘((DVecH‘𝐾)‘𝑊)) = (LFnl‘((DVecH‘𝐾)‘𝑊))
7		eqid 2734	. . . 4 ⊢ (LKer‘((DVecH‘𝐾)‘𝑊)) = (LKer‘((DVecH‘𝐾)‘𝑊))
8		eqid 2734	. . . 4 ⊢ (LDual‘((DVecH‘𝐾)‘𝑊)) = (LDual‘((DVecH‘𝐾)‘𝑊))
9		eqid 2734	. . . 4 ⊢ (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))) = (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊)))
10		eqid 2734	. . . 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 41847	. . 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 6771	. . 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 6783	. . 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 18	. 2 ⊢ (𝜑 → 𝑀:(LSubSp‘((DVecH‘𝐾)‘𝑊))–1-1-onto→ran 𝑀)
16		mapdcnvid2.x	. 2 ⊢ (𝜑 → 𝑋 ∈ ran 𝑀)
17		f1ocnvfv2 7221	. 2 ⊢ ((𝑀:(LSubSp‘((DVecH‘𝐾)‘𝑊))–1-1-onto→ran 𝑀 ∧ 𝑋 ∈ ran 𝑀) → (𝑀‘(^◡𝑀‘𝑋)) = 𝑋)
18	15, 16, 17	syl2anc 584	1 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 {crab 3397 ∩ cin 3898 𝒫 cpw 4552 ^◡ccnv 5621 ran crn 5623 –1-1→wf1 6487 –1-1-onto→wf1o 6489 ‘cfv 6490 LSubSpclss 20880 LFnlclfn 39256 LKerclk 39284 LDualcld 39322 HLchlt 39549 LHypclh 40183 DVecHcdvh 41277 ocHcoch 41546 mapdcmpd 41823

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678 ax-cnex 11080 ax-resscn 11081 ax-1cn 11082 ax-icn 11083 ax-addcl 11084 ax-addrcl 11085 ax-mulcl 11086 ax-mulrcl 11087 ax-mulcom 11088 ax-addass 11089 ax-mulass 11090 ax-distr 11091 ax-i2m1 11092 ax-1ne0 11093 ax-1rid 11094 ax-rnegex 11095 ax-rrecex 11096 ax-cnre 11097 ax-pre-lttri 11098 ax-pre-lttrn 11099 ax-pre-ltadd 11100 ax-pre-mulgt0 11101 ax-riotaBAD 39152

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-nel 3035 df-ral 3050 df-rex 3059 df-rmo 3348 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-tp 4583 df-op 4585 df-uni 4862 df-int 4901 df-iun 4946 df-iin 4947 df-br 5097 df-opab 5159 df-mpt 5178 df-tr 5204 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-undef 8213 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-er 8633 df-map 8763 df-en 8882 df-dom 8883 df-sdom 8884 df-fin 8885 df-pnf 11166 df-mnf 11167 df-xr 11168 df-ltxr 11169 df-le 11170 df-sub 11364 df-neg 11365 df-nn 12144 df-2 12206 df-3 12207 df-4 12208 df-5 12209 df-6 12210 df-n0 12400 df-z 12487 df-uz 12750 df-fz 13422 df-struct 17072 df-sets 17089 df-slot 17107 df-ndx 17119 df-base 17135 df-ress 17156 df-plusg 17188 df-mulr 17189 df-sca 17191 df-vsca 17192 df-0g 17359 df-mre 17503 df-mrc 17504 df-acs 17506 df-proset 18215 df-poset 18234 df-plt 18249 df-lub 18265 df-glb 18266 df-join 18267 df-meet 18268 df-p0 18344 df-p1 18345 df-lat 18353 df-clat 18420 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-submnd 18707 df-grp 18864 df-minusg 18865 df-sbg 18866 df-subg 19051 df-cntz 19244 df-oppg 19273 df-lsm 19563 df-cmn 19709 df-abl 19710 df-mgp 20074 df-rng 20086 df-ur 20115 df-ring 20168 df-oppr 20271 df-dvdsr 20291 df-unit 20292 df-invr 20322 df-dvr 20335 df-nzr 20444 df-rlreg 20625 df-domn 20626 df-drng 20662 df-lmod 20811 df-lss 20881 df-lsp 20921 df-lvec 21053 df-lsatoms 39175 df-lshyp 39176 df-lcv 39218 df-lfl 39257 df-lkr 39285 df-ldual 39323 df-oposet 39375 df-ol 39377 df-oml 39378 df-covers 39465 df-ats 39466 df-atl 39497 df-cvlat 39521 df-hlat 39550 df-llines 39697 df-lplanes 39698 df-lvols 39699 df-lines 39700 df-psubsp 39702 df-pmap 39703 df-padd 39995 df-lhyp 40187 df-laut 40188 df-ldil 40303 df-ltrn 40304 df-trl 40358 df-tgrp 40942 df-tendo 40954 df-edring 40956 df-dveca 41202 df-disoa 41228 df-dvech 41278 df-dib 41338 df-dic 41372 df-dih 41428 df-doch 41547 df-djh 41594 df-mapd 41824

This theorem is referenced by: mapdcnvordN 41857 mapdcv 41859 mapdin 41861 mapdlsm 41863 mapdcnvatN 41865 hdmaprnlem3N 42049 hdmaprnlem9N 42056 hdmaprnlem16N 42061

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