dia2dimlem6 - Mathbox for Norm Megill

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Theorem dia2dimlem6 41863

Description: Lemma for dia2dim 41871. Eliminate auxiliary translations 𝐺 and 𝐷. (Contributed by NM, 8-Sep-2014.)

**Hypotheses**
Ref	Expression
dia2dimlem6.l	⊢ ≤ = (le‘𝐾)
dia2dimlem6.j	⊢ ∨ = (join‘𝐾)
dia2dimlem6.m	⊢ ∧ = (meet‘𝐾)
dia2dimlem6.a	⊢ 𝐴 = (Atoms‘𝐾)
dia2dimlem6.h	⊢ 𝐻 = (LHyp‘𝐾)
dia2dimlem6.t	⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
dia2dimlem6.r	⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
dia2dimlem6.y	⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
dia2dimlem6.s	⊢ 𝑆 = (LSubSp‘𝑌)
dia2dimlem6.pl	⊢ ⊕ = (LSSum‘𝑌)
dia2dimlem6.n	⊢ 𝑁 = (LSpan‘𝑌)
dia2dimlem6.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
dia2dimlem6.q	⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))
dia2dimlem6.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dia2dimlem6.u	⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
dia2dimlem6.v	⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
dia2dimlem6.p	⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
dia2dimlem6.f	⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
dia2dimlem6.rf	⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
dia2dimlem6.uv	⊢ (𝜑 → 𝑈 ≠ 𝑉)
dia2dimlem6.ru	⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
dia2dimlem6.rv	⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)

**Assertion**
Ref	Expression
dia2dimlem6	⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

Proof of Theorem dia2dimlem6 Dummy variables 𝑔 𝑑 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dia2dimlem6.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
2		dia2dimlem6.l	. . . 4 ⊢ ≤ = (le‘𝐾)
3		dia2dimlem6.j	. . . 4 ⊢ ∨ = (join‘𝐾)
4		dia2dimlem6.m	. . . 4 ⊢ ∧ = (meet‘𝐾)
5		dia2dimlem6.a	. . . 4 ⊢ 𝐴 = (Atoms‘𝐾)
6		dia2dimlem6.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		dia2dimlem6.t	. . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)
8		dia2dimlem6.r	. . . 4 ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)
9		dia2dimlem6.q	. . . 4 ⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))
10		dia2dimlem6.u	. . . 4 ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
11		dia2dimlem6.v	. . . 4 ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
12		dia2dimlem6.p	. . . 4 ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
13		dia2dimlem6.f	. . . 4 ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
14		dia2dimlem6.rf	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
15		dia2dimlem6.uv	. . . 4 ⊢ (𝜑 → 𝑈 ≠ 𝑉)
16		dia2dimlem6.ru	. . . 4 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)
17	2, 3, 4, 5, 6, 7, 8, 9, 1, 10, 11, 12, 13, 14, 15, 16	dia2dimlem1 41858	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
18	13	simpld 499	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
19	2, 5, 6, 7	ltrnel 40933	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
20	1, 18, 12, 19	syl3anc 1398	. . 3 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
21	2, 5, 6, 7	cdleme50ex 41353	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊)) → ∃𝑑 ∈ 𝑇 (𝑑‘𝑄) = (𝐹‘𝑃))
22	1, 17, 20, 21	syl3anc 1398	. 2 ⊢ (𝜑 → ∃𝑑 ∈ 𝑇 (𝑑‘𝑄) = (𝐹‘𝑃))
23	2, 5, 6, 7	cdleme50ex 41353	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
24	1, 12, 17, 23	syl3anc 1398	. . . 4 ⊢ (𝜑 → ∃𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
25		dia2dimlem6.y	. . . . . . . 8 ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)
26		dia2dimlem6.s	. . . . . . . 8 ⊢ 𝑆 = (LSubSp‘𝑌)
27		dia2dimlem6.pl	. . . . . . . 8 ⊢ ⊕ = (LSSum‘𝑌)
28		dia2dimlem6.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑌)
29		dia2dimlem6.i	. . . . . . . 8 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
30	1	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	10	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
32	11	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
33	12	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
34	13	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
35	14	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
36	15	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝑈 ≠ 𝑉)
37	16	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑅‘𝐹) ≠ 𝑈)
38		dia2dimlem6.rv	. . . . . . . . 9 ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)
39	38	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑅‘𝐹) ≠ 𝑉)
40		simp21 1225	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝑔 ∈ 𝑇)
41		simp22 1226	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑔‘𝑃) = 𝑄)
42		simp23 1227	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝑑 ∈ 𝑇)
43		simp3 1156	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑑‘𝑄) = (𝐹‘𝑃))
44	2, 3, 4, 5, 6, 7, 8, 25, 26, 27, 28, 29, 9, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43	dia2dimlem5 41862	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
45	44	3exp 1137	. . . . . 6 ⊢ (𝜑 → ((𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) → ((𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))))
46	45	3expd 1372	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ 𝑇 → ((𝑔‘𝑃) = 𝑄 → (𝑑 ∈ 𝑇 → ((𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))))))
47	46	rexlimdv 3164	. . . 4 ⊢ (𝜑 → (∃𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄 → (𝑑 ∈ 𝑇 → ((𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))))))
48	24, 47	mpd 16	. . 3 ⊢ (𝜑 → (𝑑 ∈ 𝑇 → ((𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))))
49	48	rexlimdv 3164	. 2 ⊢ (𝜑 → (∃𝑑 ∈ 𝑇 (𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))))
50	22, 49	mpd 16	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 ∧ w3a 1103 = wceq 1570 ∈ wcel 2143 ≠ wne 2958 ∃wrex 3089 class class class wbr 5109 ‘cfv 6536 (class class class)co 7410 lecple 17312 joincjn 18362 meetcmee 18363 LSSumclsm 19699 LSubSpclss 21052 LSpanclspn 21092 Atomscatm 40057 HLchlt 40144 LHypclh 40778 LTrncltrn 40895 trLctrl 40952 DVecAcdveca 41796 DIsoAcdia 41822

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-riotaBAD 39747

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-undef 8265 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-n0 12500 df-z 12587 df-uz 12858 df-fz 13531 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-sca 17321 df-vsca 17322 df-0g 17489 df-proset 18345 df-poset 18364 df-plt 18379 df-lub 18395 df-glb 18396 df-join 18397 df-meet 18398 df-p0 18474 df-p1 18475 df-lat 18483 df-clat 18550 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-grp 18998 df-minusg 18999 df-sbg 19000 df-subg 19184 df-lsm 19701 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-drng 20829 df-lmod 20983 df-lss 21053 df-lsp 21093 df-lvec 21224 df-oposet 39970 df-ol 39972 df-oml 39973 df-covers 40060 df-ats 40061 df-atl 40092 df-cvlat 40116 df-hlat 40145 df-llines 40292 df-lplanes 40293 df-lvols 40294 df-lines 40295 df-psubsp 40297 df-pmap 40298 df-padd 40590 df-lhyp 40782 df-laut 40783 df-ldil 40898 df-ltrn 40899 df-trl 40953 df-tgrp 41537 df-tendo 41549 df-edring 41551 df-dveca 41797 df-disoa 41823

This theorem is referenced by: dia2dimlem7 41864

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