cdleme39n - Mathbox for Norm Megill

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Theorem cdleme39n 34079

Description: Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on

$.\/$

line. TODO: FIX COMMENT.

serve as f(t), f(u), f_t(

), f_t(

). Put hypotheses of cdleme38n 34077 in convention of cdleme32sn1awN 34045. TODO see if this hypothesis conversion would be better if done earlier. (Contributed by NM, 15-Mar-2013.)

**Hypotheses**
Ref	Expression
cdleme39.l
cdleme39.j	$.\/$
cdleme39.m	$./\$
cdleme39.a
cdleme39.h
cdleme39.u	$.\/$ $./\$
cdleme39.e	$.\/$ $./\$ $.\/$ $.\/$ $./\$
cdleme39.g	$.\/$ $./\$ $.\/$ $.\/$ $./\$
cdleme39.y	$.\/$ $./\$ $.\/$ $.\/$ $./\$
cdleme39.z	$.\/$ $./\$ $.\/$ $.\/$ $./\$

**Assertion**
Ref	Expression
cdleme39n	$/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$

**Proof of Theorem cdleme39n**
Step	Hyp	Ref	Expression
1		cdleme39.l	. . 3
2		cdleme39.j	. . 3 $.\/$
3		cdleme39.m	. . 3 $./\$
4		cdleme39.a	. . 3
5		cdleme39.h	. . 3
6		cdleme39.u	. . 3 $.\/$ $./\$
7		cdleme39.e	. . 3 $.\/$ $./\$ $.\/$ $.\/$ $./\$
8		cdleme39.y	. . 3 $.\/$ $./\$ $.\/$ $.\/$ $./\$
9		eqid 2462	. . 3 $.\/$ $./\$ $.\/$ $./\$
10		eqid 2462	. . 3 $.\/$ $./\$ $.\/$ $./\$
11		eqid 2462	. . 3 $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$
12		eqid 2462	. . 3 $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	cdleme38n 34077	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$
14		simp11 1044	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$
15		simp12l 1127	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
16		simp13l 1129	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
17		simp22l 1133	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
18		simp22r 1134	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
19		simp311 1161	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
20		simp32l 1139	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$
21		cdleme39.g	. . . 4 $.\/$ $./\$ $.\/$ $.\/$ $./\$
22	1, 2, 3, 4, 5, 6, 7, 21, 9	cdleme39a 34078	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$
23	14, 15, 16, 17, 18, 19, 20, 22	syl322anc 1304	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$
24		simp23l 1135	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
25		simp23r 1136	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$
26		simp312 1162	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$
27		simp33l 1141	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $/\$
28		cdleme39.z	. . . 4 $.\/$ $./\$ $.\/$ $.\/$ $./\$
29	1, 2, 3, 4, 5, 6, 8, 28, 10	cdleme39a 34078	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$
30	14, 15, 16, 24, 25, 26, 27, 29	syl322anc 1304	. 2 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$ $.\/$ $.\/$ $./\$ $./\$ $.\/$ $.\/$ $./\$
31	13, 23, 30	3netr4d 2713	1 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $.\/$ $/\$ $/\$ $/\$ $/\$ $.\/$ $/\$ $/\$ $/\$ $.\/$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 375 $/\$ w3a 991

wceq 1455

wcel 1898

wne 2633 class class class wbr 4418

cfv 5605 (class class class)co 6320

cple 15252

cjn 16244

cmee 16245

catm 32875

chlt 32962

clh 33595

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-rep 4531 ax-sep 4541 ax-nul 4550 ax-pow 4598 ax-pr 4656 ax-un 6615

This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-iun 4294 df-iin 4295 df-br 4419 df-opab 4478 df-mpt 4479 df-id 4771 df-xp 4862 df-rel 4863 df-cnv 4864 df-co 4865 df-dm 4866 df-rn 4867 df-res 4868 df-ima 4869 df-iota 5569 df-fun 5607 df-fn 5608 df-f 5609 df-f1 5610 df-fo 5611 df-f1o 5612 df-fv 5613 df-riota 6282 df-ov 6323 df-oprab 6324 df-mpt2 6325 df-1st 6825 df-2nd 6826 df-preset 16228 df-poset 16246 df-plt 16259 df-lub 16275 df-glb 16276 df-join 16277 df-meet 16278 df-p0 16340 df-p1 16341 df-lat 16347 df-clat 16409 df-oposet 32788 df-ol 32790 df-oml 32791 df-covers 32878 df-ats 32879 df-atl 32910 df-cvlat 32934 df-hlat 32963 df-llines 33109 df-lplanes 33110 df-lvols 33111 df-lines 33112 df-psubsp 33114 df-pmap 33115 df-padd 33407 df-lhyp 33599

This theorem is referenced by: cdleme40m 34080

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