mapdordlem1a - Mathbox for Norm Megill

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Theorem mapdordlem1a 42080

Description: Lemma for mapdord 42084. (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 773	. . . . . 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 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → 𝐽 ∈ 𝐹)
11	2, 3, 4, 5, 6, 7, 9, 10	dochlkr 41831	. . . . . 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 232	. . . . 5 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → ((𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽) ∧ (𝐿‘𝐽) ∈ 𝑌))
13	12	simpld 494	. . . 4 ⊢ ((𝜑 ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)) → (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽))
14	13	ex 412	. . 3 ⊢ (𝜑 → ((𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) → (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)))
15	14	pm4.71rd 562	. 2 ⊢ (𝜑 → ((𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) ↔ ((𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽) ∧ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))))
16		2fveq3 6845	. . . . 5 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝐿‘𝑔)) = (𝑂‘(𝐿‘𝐽)))
17	16	fveq2d 6844	. . . 4 ⊢ (𝑔 = 𝐽 → (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝑂‘(𝑂‘(𝐿‘𝐽))))
18	17	eleq1d 2821	. . 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 3637	. 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 41937	. . . 4 ⊢ (𝐽 ∈ 𝐶 ↔ (𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)))
23	22	anbi1i 625	. . 3 ⊢ ((𝐽 ∈ 𝐶 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌) ↔ ((𝐽 ∈ 𝐹 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) = (𝐿‘𝐽)) ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌))
24		anass 468	. . 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 646	. . 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 297	. 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 314	1 ⊢ (𝜑 → (𝐽 ∈ 𝑇 ↔ (𝐽 ∈ 𝐶 ∧ (𝑂‘(𝑂‘(𝐿‘𝐽))) ∈ 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 {crab 3389 ‘cfv 6498 Basecbs 17179 LSHypclsh 39421 LFnlclfn 39503 LKerclk 39531 HLchlt 39796 LHypclh 40430 DVecHcdvh 41524 ocHcoch 41793

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5307 ax-pr 5375 ax-un 7689 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39399

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3062 df-rmo 3342 df-reu 3343 df-rab 3390 df-v 3431 df-sbc 3729 df-csb 3838 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-pss 3909 df-nul 4274 df-if 4467 df-pw 4543 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4851 df-int 4890 df-iun 4935 df-iin 4936 df-br 5086 df-opab 5148 df-mpt 5167 df-tr 5193 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6265 df-ord 6326 df-on 6327 df-lim 6328 df-suc 6329 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-f1 6503 df-fo 6504 df-f1o 6505 df-fv 6506 df-riota 7324 df-ov 7370 df-oprab 7371 df-mpo 7372 df-om 7818 df-1st 7942 df-2nd 7943 df-tpos 8176 df-undef 8223 df-frecs 8231 df-wrecs 8262 df-recs 8311 df-rdg 8349 df-1o 8405 df-er 8643 df-map 8775 df-en 8894 df-dom 8895 df-sdom 8896 df-fin 8897 df-pnf 11181 df-mnf 11182 df-xr 11183 df-ltxr 11184 df-le 11185 df-sub 11379 df-neg 11380 df-nn 12175 df-2 12244 df-3 12245 df-4 12246 df-5 12247 df-6 12248 df-n0 12438 df-z 12525 df-uz 12789 df-fz 13462 df-struct 17117 df-sets 17134 df-slot 17152 df-ndx 17164 df-base 17180 df-ress 17201 df-plusg 17233 df-mulr 17234 df-sca 17236 df-vsca 17237 df-0g 17404 df-proset 18260 df-poset 18279 df-plt 18294 df-lub 18310 df-glb 18311 df-join 18312 df-meet 18313 df-p0 18389 df-p1 18390 df-lat 18398 df-clat 18465 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18752 df-grp 18912 df-minusg 18913 df-sbg 18914 df-subg 19099 df-cntz 19292 df-lsm 19611 df-cmn 19757 df-abl 19758 df-mgp 20122 df-rng 20134 df-ur 20163 df-ring 20216 df-oppr 20317 df-dvdsr 20337 df-unit 20338 df-invr 20368 df-dvr 20381 df-drng 20708 df-lmod 20857 df-lss 20927 df-lsp 20967 df-lvec 21098 df-lsatoms 39422 df-lshyp 39423 df-lfl 39504 df-lkr 39532 df-oposet 39622 df-ol 39624 df-oml 39625 df-covers 39712 df-ats 39713 df-atl 39744 df-cvlat 39768 df-hlat 39797 df-llines 39944 df-lplanes 39945 df-lvols 39946 df-lines 39947 df-psubsp 39949 df-pmap 39950 df-padd 40242 df-lhyp 40434 df-laut 40435 df-ldil 40550 df-ltrn 40551 df-trl 40605 df-tendo 41201 df-edring 41203 df-disoa 41475 df-dvech 41525 df-dib 41585 df-dic 41619 df-dih 41675 df-doch 41794

This theorem is referenced by: mapdordlem2 42083

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