dia2dimlem6 - Mathbox for Norm Megill

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Theorem dia2dimlem6 40674

Description: Lemma for dia2dim 40682. 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 40669	. . 3 ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
18	13	simpld 493	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑇)
19	2, 5, 6, 7	ltrnel 39744	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
20	1, 18, 12, 19	syl3anc 1368	. . 3 ⊢ (𝜑 → ((𝐹‘𝑃) ∈ 𝐴 ∧ ¬ (𝐹‘𝑃) ≤ 𝑊))
21	2, 5, 6, 7	cdleme50ex 40164	. . 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 1368	. 2 ⊢ (𝜑 → ∃𝑑 ∈ 𝑇 (𝑑‘𝑄) = (𝐹‘𝑃))
23	2, 5, 6, 7	cdleme50ex 40164	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ∃𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄)
24	1, 12, 17, 23	syl3anc 1368	. . . 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 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	10	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))
32	11	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))
33	12	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))
34	13	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))
35	14	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
36	15	3ad2ant1 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝑈 ≠ 𝑉)
37	16	3ad2ant1 1130	. . . . . . . 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 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑅‘𝐹) ≠ 𝑉)
40		simp21 1203	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝑔 ∈ 𝑇)
41		simp22 1204	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → (𝑔‘𝑃) = 𝑄)
42		simp23 1205	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝑑 ∈ 𝑇)
43		simp3 1135	. . . . . . . 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 40673	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) ∧ (𝑑‘𝑄) = (𝐹‘𝑃)) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
45	44	3exp 1116	. . . . . 6 ⊢ (𝜑 → ((𝑔 ∈ 𝑇 ∧ (𝑔‘𝑃) = 𝑄 ∧ 𝑑 ∈ 𝑇) → ((𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))))
46	45	3expd 1350	. . . . 5 ⊢ (𝜑 → (𝑔 ∈ 𝑇 → ((𝑔‘𝑃) = 𝑄 → (𝑑 ∈ 𝑇 → ((𝑑‘𝑄) = (𝐹‘𝑃) → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))))))
47	46	rexlimdv 3142	. . . 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 3142	. 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∃wrex 3059 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 lecple 17248 joincjn 18311 meetcmee 18312 LSSumclsm 19606 LSubSpclss 20832 LSpanclspn 20872 Atomscatm 38867 HLchlt 38954 LHypclh 39589 LTrncltrn 39706 trLctrl 39763 DVecAcdveca 40607 DIsoAcdia 40633

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-riotaBAD 38557

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-sca 17257 df-vsca 17258 df-0g 17431 df-proset 18295 df-poset 18313 df-plt 18330 df-lub 18346 df-glb 18347 df-join 18348 df-meet 18349 df-p0 18425 df-p1 18426 df-lat 18432 df-clat 18499 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19091 df-lsm 19608 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-ring 20192 df-oppr 20290 df-dvdsr 20313 df-unit 20314 df-invr 20344 df-dvr 20357 df-drng 20643 df-lmod 20762 df-lss 20833 df-lsp 20873 df-lvec 21005 df-oposet 38780 df-ol 38782 df-oml 38783 df-covers 38870 df-ats 38871 df-atl 38902 df-cvlat 38926 df-hlat 38955 df-llines 39103 df-lplanes 39104 df-lvols 39105 df-lines 39106 df-psubsp 39108 df-pmap 39109 df-padd 39401 df-lhyp 39593 df-laut 39594 df-ldil 39709 df-ltrn 39710 df-trl 39764 df-tgrp 40348 df-tendo 40360 df-edring 40362 df-dveca 40608 df-disoa 40634

This theorem is referenced by: dia2dimlem7 40675

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