adderpqlem - Metamath Proof Explorer

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Theorem adderpqlem 9776

Description: Lemma for adderpq 9778. (Contributed by Mario Carneiro, 8-May-2013.) (New usage is discouraged.)

**Assertion**
Ref	Expression
adderpqlem	$/\$ $/\$

Proof of Theorem adderpqlem Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xp1st 7198	. . . . . 6
2	1	3ad2ant1 1082	. . . . 5 $/\$ $/\$
3		xp2nd 7199	. . . . . 6
4	3	3ad2ant3 1084	. . . . 5 $/\$ $/\$
5		mulclpi 9715	. . . . 5 $/\$
6	2, 4, 5	syl2anc 693	. . . 4 $/\$ $/\$
7		xp1st 7198	. . . . . 6
8	7	3ad2ant3 1084	. . . . 5 $/\$ $/\$
9		xp2nd 7199	. . . . . 6
10	9	3ad2ant1 1082	. . . . 5 $/\$ $/\$
11		mulclpi 9715	. . . . 5 $/\$
12	8, 10, 11	syl2anc 693	. . . 4 $/\$ $/\$
13		addclpi 9714	. . . 4 $/\$
14	6, 12, 13	syl2anc 693	. . 3 $/\$ $/\$
15		mulclpi 9715	. . . 4 $/\$
16	10, 4, 15	syl2anc 693	. . 3 $/\$ $/\$
17		xp1st 7198	. . . . . 6
18	17	3ad2ant2 1083	. . . . 5 $/\$ $/\$
19		mulclpi 9715	. . . . 5 $/\$
20	18, 4, 19	syl2anc 693	. . . 4 $/\$ $/\$
21		xp2nd 7199	. . . . . 6
22	21	3ad2ant2 1083	. . . . 5 $/\$ $/\$
23		mulclpi 9715	. . . . 5 $/\$
24	8, 22, 23	syl2anc 693	. . . 4 $/\$ $/\$
25		addclpi 9714	. . . 4 $/\$
26	20, 24, 25	syl2anc 693	. . 3 $/\$ $/\$
27		mulclpi 9715	. . . 4 $/\$
28	22, 4, 27	syl2anc 693	. . 3 $/\$ $/\$
29		enqbreq 9741	. . 3 $/\$ $/\$ $/\$
30	14, 16, 26, 28, 29	syl22anc 1327	. 2 $/\$ $/\$
31		addpipq2 9758	. . . 4 $/\$
32	31	3adant2 1080	. . 3 $/\$ $/\$
33		addpipq2 9758	. . . 4 $/\$
34	33	3adant1 1079	. . 3 $/\$ $/\$
35	32, 34	breq12d 4666	. 2 $/\$ $/\$
36		enqbreq2 9742	. . . 4 $/\$
37	36	3adant3 1081	. . 3 $/\$ $/\$
38		mulclpi 9715	. . . . 5 $/\$
39	4, 4, 38	syl2anc 693	. . . 4 $/\$ $/\$
40		mulclpi 9715	. . . . 5 $/\$
41	2, 22, 40	syl2anc 693	. . . 4 $/\$ $/\$
42		mulcanpi 9722	. . . 4 $/\$
43	39, 41, 42	syl2anc 693	. . 3 $/\$ $/\$
44		mulcompi 9718	. . . . . . . 8
45		fvex 6201	. . . . . . . . 9
46		fvex 6201	. . . . . . . . 9
47		fvex 6201	. . . . . . . . 9
48		mulcompi 9718	. . . . . . . . 9
49		mulasspi 9719	. . . . . . . . 9
50	45, 46, 47, 48, 49, 47	caov4 6865	. . . . . . . 8
51	44, 50	eqtri 2644	. . . . . . 7
52		fvex 6201	. . . . . . . . 9
53		fvex 6201	. . . . . . . . 9
54	52, 47, 53, 48, 49, 46	caov4 6865	. . . . . . . 8
55		mulcompi 9718	. . . . . . . . 9
56		mulcompi 9718	. . . . . . . . 9
57	55, 56	oveq12i 6662	. . . . . . . 8
58	54, 57	eqtri 2644	. . . . . . 7
59	51, 58	oveq12i 6662	. . . . . 6
60		addcompi 9716	. . . . . 6
61		ovex 6678	. . . . . . 7
62		ovex 6678	. . . . . . 7
63		ovex 6678	. . . . . . 7
64		distrpi 9720	. . . . . . 7
65	61, 62, 63, 48, 64	caovdir 6868	. . . . . 6
66	59, 60, 65	3eqtr4i 2654	. . . . 5
67		addcompi 9716	. . . . . 6
68		mulasspi 9719	. . . . . . . 8
69		mulcompi 9718	. . . . . . . . . 10
70		mulasspi 9719	. . . . . . . . . . . 12
71		mulcompi 9718	. . . . . . . . . . . 12
72		mulasspi 9719	. . . . . . . . . . . 12
73	70, 71, 72	3eqtrri 2649	. . . . . . . . . . 11
74	73	oveq1i 6660	. . . . . . . . . 10
75	69, 74	eqtri 2644	. . . . . . . . 9
76		mulasspi 9719	. . . . . . . . 9
77	75, 76	eqtri 2644	. . . . . . . 8
78	68, 77	eqtri 2644	. . . . . . 7
79	78	oveq2i 6661	. . . . . 6
80		distrpi 9720	. . . . . 6
81	67, 79, 80	3eqtr4i 2654	. . . . 5
82	66, 81	eqeq12i 2636	. . . 4
83		mulclpi 9715	. . . . . 6 $/\$
84	16, 24, 83	syl2anc 693	. . . . 5 $/\$ $/\$
85		mulclpi 9715	. . . . . 6 $/\$
86	39, 41, 85	syl2anc 693	. . . . 5 $/\$ $/\$
87		addcanpi 9721	. . . . 5 $/\$
88	84, 86, 87	syl2anc 693	. . . 4 $/\$ $/\$
89	82, 88	syl5rbbr 275	. . 3 $/\$ $/\$
90	37, 43, 89	3bitr2d 296	. 2 $/\$ $/\$
91	30, 35, 90	3bitr4rd 301	1 $/\$ $/\$

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ w3a 1037

wceq 1483

wcel 1990

cop 4183 class class class wbr 4653

cxp 5112

cfv 5888 (class class class)co 6650

c1st 7166

c2nd 7167

cnpi 9666

cpli 9667

cmi 9668

cplpq 9670

ceq 9673

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

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-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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-oadd 7564 df-omul 7565 df-ni 9694 df-pli 9695 df-mi 9696 df-plpq 9730 df-enq 9733

This theorem is referenced by: adderpq 9778

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