mapdrvallem3 - Mathbox for Norm Megill

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Theorem mapdrvallem3 42138

Description: Lemma for mapdrval 42139. (Contributed by NM, 2-Feb-2015.)

**Hypotheses**
Ref	Expression
mapdrval.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdrval.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
mapdrval.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdrval.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdrval.s	⊢ 𝑆 = (LSubSp‘𝑈)
mapdrval.f	⊢ 𝐹 = (LFnl‘𝑈)
mapdrval.l	⊢ 𝐿 = (LKer‘𝑈)
mapdrval.d	⊢ 𝐷 = (LDual‘𝑈)
mapdrval.t	⊢ 𝑇 = (LSubSp‘𝐷)
mapdrval.c	⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
mapdrval.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdrval.r	⊢ (𝜑 → 𝑅 ∈ 𝑇)
mapdrval.e	⊢ (𝜑 → 𝑅 ⊆ 𝐶)
mapdrval.q	⊢ 𝑄 = ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ))
mapdrval.v	⊢ 𝑉 = (Base‘𝑈)
mapdrvallem2.a	⊢ 𝐴 = (LSAtoms‘𝑈)
mapdrvallem2.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdrvallem2.z	⊢ 0 = (0_g‘𝑈)
mapdrvallem2.y	⊢ 𝑌 = (0_g‘𝐷)

**Assertion**
Ref	Expression
mapdrvallem3	⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} = 𝑅)

Distinct variable groups: 𝐶,𝑓 𝑓,𝑔,𝐹 𝑓,𝐾 𝑔,ℎ,𝐿 𝑔,𝑂,ℎ 𝑄,𝑓,ℎ 𝑅,𝑓,ℎ 𝑈,𝑔 𝑓,𝑊 𝜑,𝑓 𝐶,ℎ ℎ,𝑁 𝑄,ℎ 𝑈,ℎ ℎ,𝑉 ℎ,𝑌 0 ,ℎ 𝜑,ℎ Allowed substitution hints: 𝜑(𝑔) 𝐴(𝑓,𝑔,ℎ) 𝐶(𝑔) 𝐷(𝑓,𝑔,ℎ) 𝑄(𝑔) 𝑅(𝑔) 𝑆(𝑓,𝑔,ℎ) 𝑇(𝑓,𝑔,ℎ) 𝑈(𝑓) 𝐹(ℎ) 𝐻(𝑓,𝑔,ℎ) 𝐾(𝑔,ℎ) 𝐿(𝑓) 𝑀(𝑓,𝑔,ℎ) 𝑁(𝑓,𝑔) 𝑂(𝑓) 𝑉(𝑓,𝑔) 𝑊(𝑔,ℎ) 𝑌(𝑓,𝑔) 0 (𝑓,𝑔)

**Proof of Theorem mapdrvallem3**
Step	Hyp	Ref	Expression
1		mapdrval.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdrval.o	. . 3 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
3		mapdrval.m	. . 3 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
4		mapdrval.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		mapdrval.s	. . 3 ⊢ 𝑆 = (LSubSp‘𝑈)
6		mapdrval.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
7		mapdrval.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
8		mapdrval.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
9		mapdrval.t	. . 3 ⊢ 𝑇 = (LSubSp‘𝐷)
10		mapdrval.c	. . 3 ⊢ 𝐶 = {𝑔 ∈ 𝐹 ∣ (𝑂‘(𝑂‘(𝐿‘𝑔))) = (𝐿‘𝑔)}
11		mapdrval.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		mapdrval.r	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑇)
13		mapdrval.e	. . 3 ⊢ (𝜑 → 𝑅 ⊆ 𝐶)
14		mapdrval.q	. . 3 ⊢ 𝑄 = ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ))
15		mapdrval.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
16		mapdrvallem2.a	. . 3 ⊢ 𝐴 = (LSAtoms‘𝑈)
17		mapdrvallem2.n	. . 3 ⊢ 𝑁 = (LSpan‘𝑈)
18		mapdrvallem2.z	. . 3 ⊢ 0 = (0_g‘𝑈)
19		mapdrvallem2.y	. . 3 ⊢ 𝑌 = (0_g‘𝐷)
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	mapdrvallem2 42137	. 2 ⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} ⊆ 𝑅)
21		2fveq3 6832	. . . . . 6 ⊢ (ℎ = 𝑓 → (𝑂‘(𝐿‘ℎ)) = (𝑂‘(𝐿‘𝑓)))
22	21	ssiun2s 4978	. . . . 5 ⊢ (𝑓 ∈ 𝑅 → (𝑂‘(𝐿‘𝑓)) ⊆ ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ)))
23	22	adantl 482	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → (𝑂‘(𝐿‘𝑓)) ⊆ ∪ ℎ ∈ 𝑅 (𝑂‘(𝐿‘ℎ)))
24	23, 14	sseqtrrdi 3956	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑅) → (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄)
25	13, 24	ssrabdv 4004	. 2 ⊢ (𝜑 → 𝑅 ⊆ {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄})
26	20, 25	eqssd 3932	1 ⊢ (𝜑 → {𝑓 ∈ 𝐶 ∣ (𝑂‘(𝐿‘𝑓)) ⊆ 𝑄} = 𝑅)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 {crab 3391 ⊆ wss 3883 ∪ ciun 4921 ‘cfv 6485 Basecbs 17170 0_gc0g 17393 LSubSpclss 20921 LSpanclspn 20961 LSAtomsclsa 39466 LFnlclfn 39549 LKerclk 39577 LDualcld 39615 HLchlt 39842 LHypclh 40476 DVecHcdvh 41570 ocHcoch 41839 mapdcmpd 42116

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-riotaBAD 39445

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 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-er 8633 df-map 8765 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-n0 12429 df-z 12516 df-uz 12780 df-fz 13453 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-sca 17227 df-vsca 17228 df-0g 17395 df-proset 18251 df-poset 18270 df-plt 18285 df-lub 18301 df-glb 18302 df-join 18303 df-meet 18304 df-p0 18380 df-p1 18381 df-lat 18389 df-clat 18456 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-grp 18903 df-minusg 18904 df-sbg 18905 df-subg 19090 df-cntz 19283 df-lsm 19602 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-dvr 20372 df-drng 20703 df-lmod 20852 df-lss 20922 df-lsp 20962 df-lvec 21093 df-lsatoms 39468 df-lshyp 39469 df-lfl 39550 df-lkr 39578 df-ldual 39616 df-oposet 39668 df-ol 39670 df-oml 39671 df-covers 39758 df-ats 39759 df-atl 39790 df-cvlat 39814 df-hlat 39843 df-llines 39990 df-lplanes 39991 df-lvols 39992 df-lines 39993 df-psubsp 39995 df-pmap 39996 df-padd 40288 df-lhyp 40480 df-laut 40481 df-ldil 40596 df-ltrn 40597 df-trl 40651 df-tgrp 41235 df-tendo 41247 df-edring 41249 df-dveca 41495 df-disoa 41521 df-dvech 41571 df-dib 41631 df-dic 41665 df-dih 41721 df-doch 41840 df-djh 41887

This theorem is referenced by: mapdrval 42139

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