dia2dimlem6 - Mathbox for Norm Megill

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Theorem dia2dimlem6 39062

Description: Lemma for dia2dim 39070. 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 39057	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
18	13	simpld 494	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
19	2, 5, 6, 7	ltrnel 38132	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
20	1, 18, 12, 19	syl3anc 1369	. . 3 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
21	2, 5, 6, 7	cdleme50ex 38552	. . 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 1369	. 2 ⊢ (𝜑 → ∃𝑑 ∈ 𝑇 (𝑑‘𝑄) = (𝐹‘𝑃))
23	2, 5, 6, 7	cdleme50ex 38552	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
24	1, 12, 17, 23	syl3anc 1369	. . . 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 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	10	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
32	11	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
33	12	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
34	13	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
35	14	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
36	15	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝑈 ≠ 𝑉)
37	16	3ad2ant1 1131	. . . . . . . 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 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑅‘𝐹) ≠ 𝑉)
40		simp21 1204	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝑔 ∈ 𝑇)
41		simp22 1205	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑔‘𝑃) = 𝑄)
42		simp23 1206	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝑑 ∈ 𝑇)
43		simp3 1136	. . . . . . . 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 39061	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
45	44	3exp 1117	. . . . . 6 ⊢ (𝜑 → ((𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) → ((𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))))
46	45	3expd 1351	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ 𝑇 → ((𝑔‘𝑃) = 𝑄 → (𝑑 ∈ 𝑇 → ((𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))))))
47	46	rexlimdv 3213	. . . 4 ⊢ (𝜑 → (∃𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄 → (𝑑 ∈ 𝑇 → ((𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))))))
48	24, 47	mpd 15	. . 3 ⊢ (𝜑 → (𝑑 ∈ 𝑇 → ((𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))))
49	48	rexlimdv 3213	. 2 ⊢ (𝜑 → (∃𝑑 ∈ 𝑇 (𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))))
50	22, 49	mpd 15	1 ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1541 ∈ wcel 2109 ≠ wne 2944 ∃wrex 3066 class class class wbr 5078 ‘cfv 6430 (class class class)co 7268 lecple 16950 joincjn 18010 meetcmee 18011 LSSumclsm 19220 LSubSpclss 20174 LSpanclspn 20214 Atomscatm 37256 HLchlt 37343 LHypclh 37977 LTrncltrn 38094 trLctrl 38151 DVecAcdveca 38995 DIsoAcdia 39021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-rep 5213 ax-sep 5226 ax-nul 5233 ax-pow 5291 ax-pr 5355 ax-un 7579 ax-cnex 10911 ax-resscn 10912 ax-1cn 10913 ax-icn 10914 ax-addcl 10915 ax-addrcl 10916 ax-mulcl 10917 ax-mulrcl 10918 ax-mulcom 10919 ax-addass 10920 ax-mulass 10921 ax-distr 10922 ax-i2m1 10923 ax-1ne0 10924 ax-1rid 10925 ax-rnegex 10926 ax-rrecex 10927 ax-cnre 10928 ax-pre-lttri 10929 ax-pre-lttrn 10930 ax-pre-ltadd 10931 ax-pre-mulgt0 10932 ax-riotaBAD 36946

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-reu 3072 df-rmo 3073 df-rab 3074 df-v 3432 df-sbc 3720 df-csb 3837 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-pss 3910 df-nul 4262 df-if 4465 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4845 df-int 4885 df-iun 4931 df-iin 4932 df-br 5079 df-opab 5141 df-mpt 5162 df-tr 5196 df-id 5488 df-eprel 5494 df-po 5502 df-so 5503 df-fr 5543 df-we 5545 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-rn 5599 df-res 5600 df-ima 5601 df-pred 6199 df-ord 6266 df-on 6267 df-lim 6268 df-suc 6269 df-iota 6388 df-fun 6432 df-fn 6433 df-f 6434 df-f1 6435 df-fo 6436 df-f1o 6437 df-fv 6438 df-riota 7225 df-ov 7271 df-oprab 7272 df-mpo 7273 df-om 7701 df-1st 7817 df-2nd 7818 df-tpos 8026 df-undef 8073 df-frecs 8081 df-wrecs 8112 df-recs 8186 df-rdg 8225 df-1o 8281 df-er 8472 df-map 8591 df-en 8708 df-dom 8709 df-sdom 8710 df-fin 8711 df-pnf 10995 df-mnf 10996 df-xr 10997 df-ltxr 10998 df-le 10999 df-sub 11190 df-neg 11191 df-nn 11957 df-2 12019 df-3 12020 df-4 12021 df-5 12022 df-6 12023 df-n0 12217 df-z 12303 df-uz 12565 df-fz 13222 df-struct 16829 df-sets 16846 df-slot 16864 df-ndx 16876 df-base 16894 df-ress 16923 df-plusg 16956 df-mulr 16957 df-sca 16959 df-vsca 16960 df-0g 17133 df-proset 17994 df-poset 18012 df-plt 18029 df-lub 18045 df-glb 18046 df-join 18047 df-meet 18048 df-p0 18124 df-p1 18125 df-lat 18131 df-clat 18198 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561 df-minusg 18562 df-sbg 18563 df-subg 18733 df-lsm 19222 df-cmn 19369 df-abl 19370 df-mgp 19702 df-ur 19719 df-ring 19766 df-oppr 19843 df-dvdsr 19864 df-unit 19865 df-invr 19895 df-dvr 19906 df-drng 19974 df-lmod 20106 df-lss 20175 df-lsp 20215 df-lvec 20346 df-oposet 37169 df-ol 37171 df-oml 37172 df-covers 37259 df-ats 37260 df-atl 37291 df-cvlat 37315 df-hlat 37344 df-llines 37491 df-lplanes 37492 df-lvols 37493 df-lines 37494 df-psubsp 37496 df-pmap 37497 df-padd 37789 df-lhyp 37981 df-laut 37982 df-ldil 38097 df-ltrn 38098 df-trl 38152 df-tgrp 38736 df-tendo 38748 df-edring 38750 df-dveca 38996 df-disoa 39022

This theorem is referenced by: dia2dimlem7 39063

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