mapdordlem1a - Mathbox for Norm Megill

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Theorem mapdordlem1a 42466

Description: Lemma for mapdord 42470. (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 785	. . . . . 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 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		simprl 783	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → 𝐽 ∈ 𝐹)
11	2, 3, 4, 5, 6, 7, 9, 10	dochlkr 42217	. . . . . 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 500	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽))
14	13	ex 418	. . 3 ⊢ (𝜑 → ((𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) → (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)))
15	14	pm4.71rd 572	. 2 ⊢ (𝜑 → ((𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) ↔ ((𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽) ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))))
16		2fveq3 6890	. . . . 5 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝐿‘𝑔)) = (𝑂‘(𝐿‘𝐽)))
17	16	fveq2d 6889	. . . 4 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝑂‘(𝑂‘(𝐿‘𝐽))))
18	17	eleq1d 2850	. . 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 3656	. 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 42323	. . . 4 ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)))
23	22	anbi1i 636	. . 3 ⊢ ((𝐽 ∈ 𝐶 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) ↔ ((𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)) ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))
24		anass 474	. . 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 658	. . 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2146 {crab 3418 ‘cfv 6540 Basecbs 17291 LSHypclsh 39807 LFnlclfn 39889 LKerclk 39917 HLchlt 40182 LHypclh 40816 DVecHcdvh 41910 ocHcoch 42179

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 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-riotaBAD 39785

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 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-n0 12520 df-z 12607 df-uz 12879 df-fz 13552 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-sca 17348 df-vsca 17349 df-0g 17516 df-proset 18372 df-poset 18391 df-plt 18406 df-lub 18422 df-glb 18423 df-join 18424 df-meet 18425 df-p0 18501 df-p1 18502 df-lat 18510 df-clat 18577 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-grp 19047 df-minusg 19048 df-sbg 19049 df-subg 19233 df-cntz 19431 df-lsm 19750 df-cmn 19896 df-abl 19897 df-mgp 20261 df-rng 20275 df-ur 20308 df-ring 20361 df-oppr 20465 df-dvdsr 20485 df-unit 20486 df-invr 20516 df-dvr 20529 df-drng 20879 df-lmod 21033 df-lss 21103 df-lsp 21143 df-lvec 21274 df-lsatoms 39808 df-lshyp 39809 df-lfl 39890 df-lkr 39918 df-oposet 40008 df-ol 40010 df-oml 40011 df-covers 40098 df-ats 40099 df-atl 40130 df-cvlat 40154 df-hlat 40183 df-llines 40330 df-lplanes 40331 df-lvols 40332 df-lines 40333 df-psubsp 40335 df-pmap 40336 df-padd 40628 df-lhyp 40820 df-laut 40821 df-ldil 40936 df-ltrn 40937 df-trl 40991 df-tendo 41587 df-edring 41589 df-disoa 41861 df-dvech 41911 df-dib 41971 df-dic 42005 df-dih 42061 df-doch 42180

This theorem is used by: mapdordlem2 42469

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