linepsubN - Mathbox for Norm Megill

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Theorem linepsubN 35038

Description: A line is a projective subspace. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
linepsub.n
linepsub.s

**Assertion**
Ref	Expression
linepsubN	$/\$

Proof of Theorem linepsubN Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3687	. . . . . . . 8
2		sseq1 3626	. . . . . . . 8
3	1, 2	mpbiri 248	. . . . . . 7
4	3	a1i 11	. . . . . 6 $/\$ $/\$
5		eqid 2622	. . . . . . . . . 10
6		eqid 2622	. . . . . . . . . 10
7	5, 6	atbase 34576	. . . . . . . . 9
8	5, 6	atbase 34576	. . . . . . . . 9
9	7, 8	anim12i 590	. . . . . . . 8 $/\$ $/\$
10		eqid 2622	. . . . . . . . . 10
11	5, 10	latjcl 17051	. . . . . . . . 9 $/\$ $/\$
12	11	3expb 1266	. . . . . . . 8 $/\$ $/\$
13	9, 12	sylan2 491	. . . . . . 7 $/\$ $/\$
14		eleq2 2690	. . . . . . . . . . . . . . . . . . 19
15		breq1 4656	. . . . . . . . . . . . . . . . . . . . 21
16	15	elrab 3363	. . . . . . . . . . . . . . . . . . . 20 $/\$
17	5, 6	atbase 34576	. . . . . . . . . . . . . . . . . . . . 21
18	17	anim1i 592	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
19	16, 18	sylbi 207	. . . . . . . . . . . . . . . . . . 19 $/\$
20	14, 19	syl6bi 243	. . . . . . . . . . . . . . . . . 18 $/\$
21		eleq2 2690	. . . . . . . . . . . . . . . . . . 19
22		breq1 4656	. . . . . . . . . . . . . . . . . . . . 21
23	22	elrab 3363	. . . . . . . . . . . . . . . . . . . 20 $/\$
24	5, 6	atbase 34576	. . . . . . . . . . . . . . . . . . . . 21
25	24	anim1i 592	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$
26	23, 25	sylbi 207	. . . . . . . . . . . . . . . . . . 19 $/\$
27	21, 26	syl6bi 243	. . . . . . . . . . . . . . . . . 18 $/\$
28	20, 27	anim12d 586	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
29		an4 865	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	28, 29	syl6ib 241	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
31	30	imp 445	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
32	31	anim2i 593	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
33	32	anassrs 680	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	5, 6	atbase 34576	. . . . . . . . . . . . 13
35		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21
36	5, 35, 10	latjle12 17062	. . . . . . . . . . . . . . . . . . . 20 $/\$ $/\$ $/\$ $/\$
37	36	biimpd 219	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$
38	37	3exp2 1285	. . . . . . . . . . . . . . . . . 18 $/\$
39	38	impd 447	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
40	39	com23 86	. . . . . . . . . . . . . . . 16 $/\$ $/\$
41	40	imp43 621	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
42	41	adantr 481	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	5, 10	latjcl 17051	. . . . . . . . . . . . . . . . . 18 $/\$ $/\$
44	43	3expib 1268	. . . . . . . . . . . . . . . . 17 $/\$
45	5, 35	lattr 17056	. . . . . . . . . . . . . . . . . . 19 $/\$ $/\$ $/\$ $/\$
46	45	3exp2 1285	. . . . . . . . . . . . . . . . . 18 $/\$
47	46	com24 95	. . . . . . . . . . . . . . . . 17 $/\$
48	44, 47	syl5d 73	. . . . . . . . . . . . . . . 16 $/\$ $/\$
49	48	imp41 619	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$ $/\$
50	49	adantlrr 757	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	42, 50	mpan2d 710	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	33, 34, 51	syl2an 494	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
53		simpr 477	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
54	52, 53	jctild 566	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55		eleq2 2690	. . . . . . . . . . . . 13
56		breq1 4656	. . . . . . . . . . . . . 14
57	56	elrab 3363	. . . . . . . . . . . . 13 $/\$
58	55, 57	syl6bb 276	. . . . . . . . . . . 12 $/\$
59	58	ad3antlr 767	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	54, 59	sylibrd 249	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$
61	60	ralrimiva 2966	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
62	61	ralrimivva 2971	. . . . . . . 8 $/\$ $/\$
63	62	ex 450	. . . . . . 7 $/\$
64	13, 63	syldan 487	. . . . . 6 $/\$ $/\$
65	4, 64	jcad 555	. . . . 5 $/\$ $/\$ $/\$
66	65	adantld 483	. . . 4 $/\$ $/\$ $/\$ $/\$
67	66	rexlimdvva 3038	. . 3 $/\$ $/\$
68		linepsub.n	. . . 4
69	35, 10, 6, 68	isline 35025	. . 3 $/\$
70		linepsub.s	. . . 4
71	35, 10, 6, 70	ispsubsp 35031	. . 3 $/\$
72	67, 69, 71	3imtr4d 283	. 2
73	72	imp 445	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wne 2794

wral 2912

wrex 2913

crab 2916

wss 3574 class class class wbr 4653

cfv 5888 (class class class)co 6650

cbs 15857

cple 15948

cjn 16944

clat 17045

catm 34550

clines 34780

cpsubsp 34782

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-poset 16946 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-lat 17046 df-ats 34554 df-lines 34787 df-psubsp 34789

This theorem is referenced by: (None)

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