colinearalglem1 - Metamath Proof Explorer

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Theorem colinearalglem1 25786

Description: Lemma for colinearalg 25790. Expand out a multiplication. (Contributed by Scott Fenton, 24-Jun-2013.)

**Assertion**
Ref	Expression
colinearalglem1	$/\$ $/\$ $/\$ $/\$ $/\$

**Proof of Theorem colinearalglem1**
Step	Hyp	Ref	Expression
1		simpl2 1065	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
2		simpl1 1064	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
3	1, 2	subcld 10392	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
4		simpr3 1069	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
5		simpr1 1067	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
6	3, 4, 5	subdid 10486	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
7	1, 2, 4	subdird 10487	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
8	1, 2, 5	subdird 10487	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
9	7, 8	oveq12d 6668	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
10		simp2 1062	. . . . . . . 8 $/\$ $/\$
11		simp3 1063	. . . . . . . 8 $/\$ $/\$
12		mulcl 10020	. . . . . . . 8 $/\$
13	10, 11, 12	syl2an 494	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
14		simp1 1061	. . . . . . . 8 $/\$ $/\$
15		mulcl 10020	. . . . . . . 8 $/\$
16	14, 11, 15	syl2an 494	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
17	13, 16	subcld 10392	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
18		simp1 1061	. . . . . . 7 $/\$ $/\$
19		mulcl 10020	. . . . . . 7 $/\$
20	10, 18, 19	syl2an 494	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
21		mulcl 10020	. . . . . . 7 $/\$
22	14, 18, 21	syl2an 494	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
23	17, 20, 22	subsub3d 10422	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
24	17, 22, 20	addsubd 10413	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
25	9, 23, 24	3eqtrrd 2661	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
26	13, 16, 20	subsub4d 10423	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
27	26	oveq1d 6665	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
28	6, 25, 27	3eqtr2d 2662	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
29		simpr2 1068	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
30	29, 5	subcld 10392	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
31		simpl3 1066	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
32	31, 2	subcld 10392	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
33	30, 32	mulcomd 10061	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
34	32, 29, 5	subdid 10486	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
35	31, 2, 29	subdird 10487	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
36	31, 2, 5	subdird 10487	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
37	35, 36	oveq12d 6668	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
38		simp3 1063	. . . . . . . . 9 $/\$ $/\$
39		simp2 1062	. . . . . . . . 9 $/\$ $/\$
40		mulcl 10020	. . . . . . . . 9 $/\$
41	38, 39, 40	syl2an 494	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
42		mulcl 10020	. . . . . . . . 9 $/\$
43	14, 39, 42	syl2an 494	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
44	41, 43	subcld 10392	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
45		mulcl 10020	. . . . . . . 8 $/\$
46	38, 18, 45	syl2an 494	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
47	44, 46, 22	subsub3d 10422	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
48	44, 22, 46	addsubd 10413	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
49	37, 47, 48	3eqtrrd 2661	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
50	41, 43, 46	subsub4d 10423	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
51	50	oveq1d 6665	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
52	49, 51	eqtr3d 2658	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
53	33, 34, 52	3eqtrd 2660	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
54	28, 53	eqeq12d 2637	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
55	16, 20	addcld 10059	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
56	13, 55	subcld 10392	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
57	43, 46	addcld 10059	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
58	41, 57	subcld 10392	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
59	56, 58, 22	addcan2d 10240	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
60	54, 59	bitrd 268	1 $/\$ $/\$ $/\$ $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

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

wceq 1483

wcel 1990 (class class class)co 6650

cc 9934

caddc 9939

cmul 9941

cmin 10266

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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 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-nel 2898 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-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 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-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sub 10268

This theorem is referenced by: colinearalglem2 25787

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