dih1dimb2 - Mathbox for Norm Megill

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Theorem dih1dimb2 41866

Description: Isomorphism H at an atom under 𝑊. (Contributed by NM, 27-Apr-2014.)

**Hypotheses**
Ref	Expression
dih1dimb2.b	⊢ 𝐵 = (Base‘𝐾)
dih1dimb2.l	⊢ ≤ = (le‘𝐾)
dih1dimb2.a	⊢ 𝐴 = (Atoms‘𝐾)
dih1dimb2.h	⊢ 𝐻 = (LHyp‘𝐾)
dih1dimb2.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dih1dimb2.o	⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
dih1dimb2.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dih1dimb2.i	⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
dih1dimb2.n	⊢ 𝑁 = (LSpan‘𝑈)

**Assertion**
Ref	Expression
dih1dimb2	⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))

Distinct variable groups: ≤ ,𝑓 𝐴,𝑓 𝐵,ℎ 𝑓,𝐻 𝑓,ℎ,𝐾 𝑄,𝑓 𝑇,𝑓,ℎ 𝑓,𝑊,ℎ Allowed substitution hints: 𝐴(ℎ) 𝐵(𝑓) 𝑄(ℎ) 𝑈(𝑓,ℎ) 𝐻(ℎ) 𝐼(𝑓,ℎ) ≤ (ℎ) 𝑁(𝑓,ℎ) 𝑂(𝑓,ℎ)

**Proof of Theorem dih1dimb2**
Step	Hyp	Ref	Expression
1		dih1dimb2.l	. . 3 ⊢ ≤ = (le‘𝐾)
2		dih1dimb2.a	. . 3 ⊢ 𝐴 = (Atoms‘𝐾)
3		dih1dimb2.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
4		dih1dimb2.t	. . 3 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
5		eqid 2763	. . 3 ⊢ ((trL‘𝐾)‘𝑊) = ((trL‘𝐾)‘𝑊)
6	1, 2, 3, 4, 5	cdlemf 41188	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄)
7		simp3 1152	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄)
8		simp1rl 1253	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑄 ∈ 𝐴)
9	7, 8	eqeltrd 2863	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴)
10		simp1l 1212	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11		simp2 1151	. . . . . . 7 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑓 ∈ 𝑇)
12		dih1dimb2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
13	12, 2, 3, 4, 5	trlnidatb 40802	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴))
14	10, 11, 13	syl2anc 593	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝑓 ≠ ( I ↾ 𝐵) ↔ (((trL‘𝐾)‘𝑊)‘𝑓) ∈ 𝐴))
15	9, 14	mpbird 259	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → 𝑓 ≠ ( I ↾ 𝐵))
16	7	fveq2d 6872	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝐼‘𝑄))
17		dih1dimb2.o	. . . . . . . 8 ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))
18		dih1dimb2.u	. . . . . . . 8 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
19		dih1dimb2.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)
20		dih1dimb2.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
21	12, 3, 4, 5, 17, 18, 19, 20	dih1dimb 41865	. . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑓 ∈ 𝑇) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝑁‘{⟨𝑓, 𝑂⟩}))
22	10, 11, 21	syl2anc 593	. . . . . 6 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘(((trL‘𝐾)‘𝑊)‘𝑓)) = (𝑁‘{⟨𝑓, 𝑂⟩}))
23	16, 22	eqtr3d 2800	. . . . 5 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩}))
24	15, 23	jca 519	. . . 4 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇 ∧ (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄) → (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))
25	24	3expia 1135	. . 3 ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ∧ 𝑓 ∈ 𝑇) → ((((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄 → (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩}))))
26	25	reximdva 3176	. 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → (∃𝑓 ∈ 𝑇 (((trL‘𝐾)‘𝑊)‘𝑓) = 𝑄 → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩}))))
27	6, 26	mpd 15	1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) → ∃𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼‘𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1099 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 ∃wrex 3087 {csn 4583 ⟨cop 4589 class class class wbr 5101 ↦ cmpt 5182 I cid 5542 ↾ cres 5650 ‘cfv 6522 Basecbs 17246 lecple 17294 LSpanclspn 21039 Atomscatm 39888 HLchlt 39975 LHypclh 40609 LTrncltrn 40726 trLctrl 40783 DVecHcdvh 41703 DIsoHcdih 41853

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-riotaBAD 39578

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-tp 4588 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-iin 4953 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-om 7848 df-1st 7971 df-2nd 7972 df-tpos 8207 df-undef 8254 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-1o 8438 df-er 8679 df-map 8811 df-en 8929 df-dom 8930 df-sdom 8931 df-fin 8932 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-nn 12212 df-2 12281 df-3 12282 df-4 12283 df-5 12284 df-6 12285 df-n0 12483 df-z 12570 df-uz 12841 df-fz 13514 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17247 df-ress 17268 df-plusg 17300 df-mulr 17301 df-sca 17303 df-vsca 17304 df-0g 17471 df-proset 18327 df-poset 18346 df-plt 18361 df-lub 18377 df-glb 18378 df-join 18379 df-meet 18380 df-p0 18456 df-p1 18457 df-lat 18465 df-clat 18532 df-mgm 18675 df-sgrp 18754 df-mnd 18770 df-grp 18979 df-minusg 18980 df-sbg 18981 df-cmn 19823 df-abl 19824 df-mgp 20188 df-rng 20200 df-ur 20233 df-ring 20286 df-oppr 20387 df-dvdsr 20407 df-unit 20408 df-invr 20438 df-dvr 20451 df-drng 20782 df-lmod 20930 df-lss 21000 df-lsp 21040 df-lvec 21171 df-oposet 39801 df-ol 39803 df-oml 39804 df-covers 39891 df-ats 39892 df-atl 39923 df-cvlat 39947 df-hlat 39976 df-llines 40123 df-lplanes 40124 df-lvols 40125 df-lines 40126 df-psubsp 40128 df-pmap 40129 df-padd 40421 df-lhyp 40613 df-laut 40614 df-ldil 40729 df-ltrn 40730 df-trl 40784 df-tendo 41380 df-edring 41382 df-disoa 41654 df-dvech 41704 df-dib 41764 df-dih 41854

This theorem is referenced by: dihatlat 41959 dihatexv 41963

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