smumullem - Metamath Proof Explorer

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Theorem smumullem 15214

Description: Lemma for smumul 15215. (Contributed by Mario Carneiro, 22-Sep-2016.)

**Hypotheses**
Ref	Expression
smumullem.a
smumullem.b
smumullem.n

**Assertion**
Ref	Expression
smumullem	bits ..^ smul bits bits

Proof of Theorem smumullem Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smumullem.n	. 2
2		oveq2 6658	. . . . . . . . . 10 ..^ ..^
3		fzo0 12492	. . . . . . . . . 10 ..^
4	2, 3	syl6eq 2672	. . . . . . . . 9 ..^
5	4	ineq2d 3814	. . . . . . . 8 bits ..^ bits
6		in0 3968	. . . . . . . 8 bits
7	5, 6	syl6eq 2672	. . . . . . 7 bits ..^
8	7	oveq1d 6665	. . . . . 6 bits ..^ smul bits smul bits
9		bitsss 15148	. . . . . . 7 bits
10		smu02 15209	. . . . . . 7 bits smul bits
11	9, 10	ax-mp 5	. . . . . 6 smul bits
12	8, 11	syl6eq 2672	. . . . 5 bits ..^ smul bits
13		oveq2 6658	. . . . . . . . 9
14		2cn 11091	. . . . . . . . . 10
15		exp0 12864	. . . . . . . . . 10
16	14, 15	ax-mp 5	. . . . . . . . 9
17	13, 16	syl6eq 2672	. . . . . . . 8
18	17	oveq2d 6666	. . . . . . 7
19	18	oveq1d 6665	. . . . . 6
20	19	fveq2d 6195	. . . . 5 bits bits
21	12, 20	eqeq12d 2637	. . . 4 bits ..^ smul bits bits bits
22	21	imbi2d 330	. . 3 bits ..^ smul bits bits bits
23		oveq2 6658	. . . . . . 7 ..^ ..^
24	23	ineq2d 3814	. . . . . 6 bits ..^ bits ..^
25	24	oveq1d 6665	. . . . 5 bits ..^ smul bits bits ..^ smul bits
26		oveq2 6658	. . . . . . . 8
27	26	oveq2d 6666	. . . . . . 7
28	27	oveq1d 6665	. . . . . 6
29	28	fveq2d 6195	. . . . 5 bits bits
30	25, 29	eqeq12d 2637	. . . 4 bits ..^ smul bits bits bits ..^ smul bits bits
31	30	imbi2d 330	. . 3 bits ..^ smul bits bits bits ..^ smul bits bits
32		oveq2 6658	. . . . . . 7 ..^ ..^
33	32	ineq2d 3814	. . . . . 6 bits ..^ bits ..^
34	33	oveq1d 6665	. . . . 5 bits ..^ smul bits bits ..^ smul bits
35		oveq2 6658	. . . . . . . 8
36	35	oveq2d 6666	. . . . . . 7
37	36	oveq1d 6665	. . . . . 6
38	37	fveq2d 6195	. . . . 5 bits bits
39	34, 38	eqeq12d 2637	. . . 4 bits ..^ smul bits bits bits ..^ smul bits bits
40	39	imbi2d 330	. . 3 bits ..^ smul bits bits bits ..^ smul bits bits
41		oveq2 6658	. . . . . . 7 ..^ ..^
42	41	ineq2d 3814	. . . . . 6 bits ..^ bits ..^
43	42	oveq1d 6665	. . . . 5 bits ..^ smul bits bits ..^ smul bits
44		oveq2 6658	. . . . . . . 8
45	44	oveq2d 6666	. . . . . . 7
46	45	oveq1d 6665	. . . . . 6
47	46	fveq2d 6195	. . . . 5 bits bits
48	43, 47	eqeq12d 2637	. . . 4 bits ..^ smul bits bits bits ..^ smul bits bits
49	48	imbi2d 330	. . 3 bits ..^ smul bits bits bits ..^ smul bits bits
50		smumullem.a	. . . . . . . 8
51		zmod10 12686	. . . . . . . 8
52	50, 51	syl 17	. . . . . . 7
53	52	oveq1d 6665	. . . . . 6
54		smumullem.b	. . . . . . . 8
55	54	zcnd 11483	. . . . . . 7
56	55	mul02d 10234	. . . . . 6
57	53, 56	eqtrd 2656	. . . . 5
58	57	fveq2d 6195	. . . 4 bits bits
59		0bits 15161	. . . 4 bits
60	58, 59	syl6req 2673	. . 3 bits
61		oveq1 6657	. . . . . 6 bits ..^ smul bits bits bits ..^ smul bits sadd bits $/\$ bits bits sadd bits $/\$ bits
62		bitsss 15148	. . . . . . . . 9 bits
63	62	a1i 11	. . . . . . . 8 $/\$ bits
64	9	a1i 11	. . . . . . . 8 $/\$ bits
65		simpr 477	. . . . . . . 8 $/\$
66	63, 64, 65	smup1 15211	. . . . . . 7 $/\$ bits ..^ smul bits bits ..^ smul bits sadd bits $/\$ bits
67		bitsinv1lem 15163	. . . . . . . . . . . 12 $/\$ bits
68	50, 67	sylan 488	. . . . . . . . . . 11 $/\$ bits
69	68	oveq1d 6665	. . . . . . . . . 10 $/\$ bits
70	50	adantr 481	. . . . . . . . . . . . 13 $/\$
71		2nn 11185	. . . . . . . . . . . . . . 15
72	71	a1i 11	. . . . . . . . . . . . . 14 $/\$
73	72, 65	nnexpcld 13030	. . . . . . . . . . . . 13 $/\$
74	70, 73	zmodcld 12691	. . . . . . . . . . . 12 $/\$
75	74	nn0cnd 11353	. . . . . . . . . . 11 $/\$
76	73	nnnn0d 11351	. . . . . . . . . . . . 13 $/\$
77		0nn0 11307	. . . . . . . . . . . . 13
78		ifcl 4130	. . . . . . . . . . . . 13 $/\$ bits
79	76, 77, 78	sylancl 694	. . . . . . . . . . . 12 $/\$ bits
80	79	nn0cnd 11353	. . . . . . . . . . 11 $/\$ bits
81	55	adantr 481	. . . . . . . . . . 11 $/\$
82	75, 80, 81	adddird 10065	. . . . . . . . . 10 $/\$ bits bits
83	80, 81	mulcomd 10061	. . . . . . . . . . 11 $/\$ bits bits
84	83	oveq2d 6666	. . . . . . . . . 10 $/\$ bits bits
85	69, 82, 84	3eqtrd 2660	. . . . . . . . 9 $/\$ bits
86	85	fveq2d 6195	. . . . . . . 8 $/\$ bits bits bits
87	74	nn0zd 11480	. . . . . . . . . 10 $/\$
88	54	adantr 481	. . . . . . . . . 10 $/\$
89	87, 88	zmulcld 11488	. . . . . . . . 9 $/\$
90	79	nn0zd 11480	. . . . . . . . . 10 $/\$ bits
91	88, 90	zmulcld 11488	. . . . . . . . 9 $/\$ bits
92		sadadd 15189	. . . . . . . . 9 $/\$ bits bits sadd bits bits bits bits
93	89, 91, 92	syl2anc 693	. . . . . . . 8 $/\$ bits sadd bits bits bits bits
94		oveq2 6658	. . . . . . . . . . . 12 bits bits
95	94	fveq2d 6195	. . . . . . . . . . 11 bits bits bits bits
96	95	eqeq1d 2624	. . . . . . . . . 10 bits bits bits $/\$ bits bits bits bits $/\$ bits
97		oveq2 6658	. . . . . . . . . . . 12 bits bits
98	97	fveq2d 6195	. . . . . . . . . . 11 bits bits bits bits
99	98	eqeq1d 2624	. . . . . . . . . 10 bits bits bits $/\$ bits bits bits bits $/\$ bits
100		bitsshft 15197	. . . . . . . . . . . 12 $/\$ bits bits
101	54, 100	sylan 488	. . . . . . . . . . 11 $/\$ bits bits
102		ibar 525	. . . . . . . . . . . 12 bits bits bits $/\$ bits
103	102	rabbidv 3189	. . . . . . . . . . 11 bits bits bits $/\$ bits
104	101, 103	sylan9req 2677	. . . . . . . . . 10 $/\$ $/\$ bits bits bits $/\$ bits
105	81	adantr 481	. . . . . . . . . . . . 13 $/\$ $/\$ bits
106	105	mul01d 10235	. . . . . . . . . . . 12 $/\$ $/\$ bits
107	106	fveq2d 6195	. . . . . . . . . . 11 $/\$ $/\$ bits bits bits
108		simpr 477	. . . . . . . . . . . . . 14 $/\$ $/\$ bits bits
109	108	intnanrd 963	. . . . . . . . . . . . 13 $/\$ $/\$ bits bits $/\$ bits
110	109	ralrimivw 2967	. . . . . . . . . . . 12 $/\$ $/\$ bits bits $/\$ bits
111		rabeq0 3957	. . . . . . . . . . . 12 bits $/\$ bits bits $/\$ bits
112	110, 111	sylibr 224	. . . . . . . . . . 11 $/\$ $/\$ bits bits $/\$ bits
113	59, 107, 112	3eqtr4a 2682	. . . . . . . . . 10 $/\$ $/\$ bits bits bits $/\$ bits
114	96, 99, 104, 113	ifbothda 4123	. . . . . . . . 9 $/\$ bits bits bits $/\$ bits
115	114	oveq2d 6666	. . . . . . . 8 $/\$ bits sadd bits bits bits sadd bits $/\$ bits
116	86, 93, 115	3eqtr2d 2662	. . . . . . 7 $/\$ bits bits sadd bits $/\$ bits
117	66, 116	eqeq12d 2637	. . . . . 6 $/\$ bits ..^ smul bits bits bits ..^ smul bits sadd bits $/\$ bits bits sadd bits $/\$ bits
118	61, 117	syl5ibr 236	. . . . 5 $/\$ bits ..^ smul bits bits bits ..^ smul bits bits
119	118	expcom 451	. . . 4 bits ..^ smul bits bits bits ..^ smul bits bits
120	119	a2d 29	. . 3 bits ..^ smul bits bits bits ..^ smul bits bits
121	22, 31, 40, 49, 60, 120	nn0ind 11472	. 2 bits ..^ smul bits bits
122	1, 121	mpcom 38	1 bits ..^ smul bits bits

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

crab 2916

cin 3573

wss 3574

c0 3915

cif 4086

cfv 5888 (class class class)co 6650

cc 9934

cc0 9936

c1 9937

caddc 9939

cmul 9941

cmin 10266

cn 11020

c2 11070

cn0 11292

cz 11377 ..^cfzo 12465

cmo 12668

cexp 12860 bitscbits 15141 sadd csad 15142 smul csmu 15143

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 ax-inf2 8538 ax-cnex 9992 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 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-xor 1465 df-tru 1486 df-fal 1489 df-had 1533 df-cad 1546 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-rmo 2920 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-int 4476 df-iun 4522 df-disj 4621 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-se 5074 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-isom 5897 df-riota 6611 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-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-xnn0 11364 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-dvds 14984 df-bits 15144 df-sad 15173 df-smu 15198

This theorem is referenced by: smumul 15215

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