mapdordlem1a - Mathbox for Norm Megill

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Theorem mapdordlem1a 42297

Description: Lemma for mapdord 42301. (Contributed by NM, 27-Jan-2015.)

**Hypotheses**
Ref	Expression
mapdordlem1a.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdordlem1a.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
mapdordlem1a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdordlem1a.v	⊢ 𝑉 = (Base‘𝑈)
mapdordlem1a.y	⊢ 𝑌 = (LSHyp‘𝑈)
mapdordlem1a.f	⊢ 𝐹 = (LFnl‘𝑈)
mapdordlem1a.l	⊢ 𝐿 = (LKer‘𝑈)
mapdordlem1a.t	⊢ 𝑇 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌}
mapdordlem1a.c	⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
mapdordlem1a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))

**Assertion**
Ref	Expression
mapdordlem1a	⊢ (𝜑 → (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐶 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)))

Distinct variable groups: 𝑔,𝐹 𝑔,𝐽 𝑔,𝐿 𝑔,𝑂 𝑔,𝑌 Allowed substitution hints: 𝜑(𝑔) 𝐶(𝑔) 𝑇(𝑔) 𝑈(𝑔) 𝐻(𝑔) 𝐾(𝑔) 𝑉(𝑔) 𝑊(𝑔)

**Proof of Theorem mapdordlem1a**
Step	Hyp	Ref	Expression
1		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)
2		mapdordlem1a.h	. . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾)
3		mapdordlem1a.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
4		mapdordlem1a.u	. . . . . . 7 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		mapdordlem1a.f	. . . . . . 7 ⊢ 𝐹 = (LFnl‘𝑈)
6		mapdordlem1a.y	. . . . . . 7 ⊢ 𝑌 = (LSHyp‘𝑈)
7		mapdordlem1a.l	. . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈)
8		mapdordlem1a.k	. . . . . . . 8 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	8	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		simprl 782	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → 𝐽 ∈ 𝐹)
11	2, 3, 4, 5, 6, 7, 9, 10	dochlkr 42048	. . . . . 6 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → ((𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌 ↔ ((𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽) ∧ (𝐿‘𝐽) ∈ 𝑌)))
12	1, 11	mpbid 235	. . . . 5 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → ((𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽) ∧ (𝐿‘𝐽) ∈ 𝑌))
13	12	simpld 499	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽))
14	13	ex 417	. . 3 ⊢ (𝜑 → ((𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) → (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)))
15	14	pm4.71rd 571	. 2 ⊢ (𝜑 → ((𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) ↔ ((𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽) ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))))
16		2fveq3 6887	. . . . 5 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝐿‘𝑔)) = (𝑂‘(𝐿‘𝐽)))
17	16	fveq2d 6886	. . . 4 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝑂‘(𝑂‘(𝐿‘𝐽))))
18	17	eleq1d 2854	. . 3 ⊢ (𝑔 = 𝐽 → ((𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌 ↔ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))
19		mapdordlem1a.t	. . 3 ⊢ 𝑇 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) ∈ 𝑌}
20	18, 19	elrab2 3663	. 2 ⊢ (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))
21		mapdordlem1a.c	. . . . 5 ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
22	21	lcfl1lem 42154	. . . 4 ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)))
23	22	anbi1i 635	. . 3 ⊢ ((𝐽 ∈ 𝐶 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) ↔ ((𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)) ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))
24		anass 473	. . 3 ⊢ (((𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)) ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) ↔ (𝐽 ∈ 𝐹 ∧ ((𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽) ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)))
25		an12 657	. . 3 ⊢ ((𝐽 ∈ 𝐹 ∧ ((𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽) ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) ↔ ((𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽) ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)))
26	23, 24, 25	3bitri 300	. 2 ⊢ ((𝐽 ∈ 𝐶 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) ↔ ((𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽) ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)))
27	15, 20, 26	3bitr4g 317	1 ⊢ (𝜑 → (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐶 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 {crab 3423 ‘cfv 6537 Basecbs 17268 LSHypclsh 39638 LFnlclfn 39720 LKerclk 39748 HLchlt 40013 LHypclh 40647 DVecHcdvh 41741 ocHcoch 42010

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-riotaBAD 39616

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 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 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7862 df-1st 7985 df-2nd 7986 df-tpos 8221 df-undef 8268 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-n0 12504 df-z 12591 df-uz 12862 df-fz 13535 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-sca 17325 df-vsca 17326 df-0g 17493 df-proset 18349 df-poset 18368 df-plt 18383 df-lub 18399 df-glb 18400 df-join 18401 df-meet 18402 df-p0 18478 df-p1 18479 df-lat 18487 df-clat 18554 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-grp 19002 df-minusg 19003 df-sbg 19004 df-subg 19188 df-cntz 19386 df-lsm 19705 df-cmn 19851 df-abl 19852 df-mgp 20216 df-rng 20230 df-ur 20263 df-ring 20316 df-oppr 20418 df-dvdsr 20438 df-unit 20439 df-invr 20469 df-dvr 20482 df-drng 20814 df-lmod 20960 df-lss 21030 df-lsp 21070 df-lvec 21201 df-lsatoms 39639 df-lshyp 39640 df-lfl 39721 df-lkr 39749 df-oposet 39839 df-ol 39841 df-oml 39842 df-covers 39929 df-ats 39930 df-atl 39961 df-cvlat 39985 df-hlat 40014 df-llines 40161 df-lplanes 40162 df-lvols 40163 df-lines 40164 df-psubsp 40166 df-pmap 40167 df-padd 40459 df-lhyp 40651 df-laut 40652 df-ldil 40767 df-ltrn 40768 df-trl 40822 df-tendo 41418 df-edring 41420 df-disoa 41692 df-dvech 41742 df-dib 41802 df-dic 41836 df-dih 41892 df-doch 42011

This theorem is referenced by: mapdordlem2 42300

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