mapdcnvid2 - Mathbox for Norm Megill

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Theorem mapdcnvid2 42714

Description: Value of the converse of the map defined by df-mapd 42682. (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 2761	. . . 4 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
3		mapdcnvid2.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
4		eqid 2761	. . . 4 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊)
5		eqid 2761	. . . 4 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊))
6		eqid 2761	. . . 4 ⊢ (LFnl‘((DVecH‘𝐾)‘𝑊)) = (LFnl‘((DVecH‘𝐾)‘𝑊))
7		eqid 2761	. . . 4 ⊢ (LKer‘((DVecH‘𝐾)‘𝑊)) = (LKer‘((DVecH‘𝐾)‘𝑊))
8		eqid 2761	. . . 4 ⊢ (LDual‘((DVecH‘𝐾)‘𝑊)) = (LDual‘((DVecH‘𝐾)‘𝑊))
9		eqid 2761	. . . 4 ⊢ (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊))) = (LSubSp‘(LDual‘((DVecH‘𝐾)‘𝑊)))
10		eqid 2761	. . . 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 42705	. . 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 6823	. . 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 6836	. . 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 7285	. 2 ⊢ ((𝑀:(LSubSp‘((DVecH‘𝐾)‘𝑊))–1-1-onto→ran 𝑀 ∧ 𝑋 ∈ ran 𝑀) → (𝑀‘(^◡𝑀‘𝑋)) = 𝑋)
18	15, 16, 17	syl2anc 596	1 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘𝑋)) = 𝑋)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 {crab 3413 ∩ cin 3898 𝒫 cpw 4557 ^◡ccnv 5650 ran crn 5652 –1-1→wf1 6535 –1-1-onto→wf1o 6537 ‘cfv 6538 LSubSpclss 21206 LFnlclfn 40114 LKerclk 40142 LDualcld 40180 HLchlt 40407 LHypclh 41041 DVecHcdvh 42135 ocHcoch 42404 mapdcmpd 42681

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 7751 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-riotaBAD 40010

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-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 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-tpos 8243 df-undef 8290 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-n0 12607 df-z 12694 df-uz 12966 df-fz 13640 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-sca 17444 df-vsca 17445 df-0g 17612 df-mre 17756 df-mrc 17757 df-acs 17759 df-proset 18468 df-poset 18487 df-plt 18502 df-lub 18518 df-glb 18519 df-join 18520 df-meet 18521 df-p0 18597 df-p1 18598 df-lat 18606 df-clat 18673 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-grp 19147 df-minusg 19148 df-sbg 19149 df-subg 19333 df-cntz 19531 df-oppg 19560 df-lsm 19850 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-nzr 20763 df-rlreg 20946 df-domn 20947 df-drng 20982 df-lmod 21137 df-lss 21207 df-lsp 21247 df-lvec 21378 df-lsatoms 40033 df-lshyp 40034 df-lcv 40076 df-lfl 40115 df-lkr 40143 df-ldual 40181 df-oposet 40233 df-ol 40235 df-oml 40236 df-covers 40323 df-ats 40324 df-atl 40355 df-cvlat 40379 df-hlat 40408 df-llines 40555 df-lplanes 40556 df-lvols 40557 df-lines 40558 df-psubsp 40560 df-pmap 40561 df-padd 40853 df-lhyp 41045 df-laut 41046 df-ldil 41161 df-ltrn 41162 df-trl 41216 df-tgrp 41800 df-tendo 41812 df-edring 41814 df-dveca 42060 df-disoa 42086 df-dvech 42136 df-dib 42196 df-dic 42230 df-dih 42286 df-doch 42405 df-djh 42452 df-mapd 42682

This theorem is used by: mapdcnvordN 42715 mapdcv 42717 mapdin 42719 mapdlsm 42721 mapdcnvatN 42723 hdmaprnlem3N 42907 hdmaprnlem9N 42914 hdmaprnlem16N 42919

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