sadaddlem - Metamath Proof Explorer

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Theorem sadaddlem 15188

Description: Lemma for sadadd 15189. (Contributed by Mario Carneiro, 9-Sep-2016.)

**Hypotheses**
Ref	Expression
sadaddlem.c	cadd bits bits
sadaddlem.k	bits
sadaddlem.1
sadaddlem.2
sadaddlem.3

**Assertion**
Ref	Expression
sadaddlem	bits sadd bits ..^ bits

Distinct variable groups:

Allowed substitution hints:

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**Proof of Theorem sadaddlem**
Step	Hyp	Ref	Expression
1		sadaddlem.k	. . . . . . . . . . . . 13 bits
2	1	fveq1i 6192	. . . . . . . . . . . 12 bits ..^ bits bits ..^
3		sadaddlem.1	. . . . . . . . . . . . . . . 16
4		2nn 11185	. . . . . . . . . . . . . . . . . 18
5	4	a1i 11	. . . . . . . . . . . . . . . . 17
6		sadaddlem.3	. . . . . . . . . . . . . . . . 17
7	5, 6	nnexpcld 13030	. . . . . . . . . . . . . . . 16
8	3, 7	zmodcld 12691	. . . . . . . . . . . . . . 15
9		fvres 6207	. . . . . . . . . . . . . . 15 bits bits
10	8, 9	syl 17	. . . . . . . . . . . . . 14 bits bits
11		bitsmod 15158	. . . . . . . . . . . . . . 15 $/\$ bits bits ..^
12	3, 6, 11	syl2anc 693	. . . . . . . . . . . . . 14 bits bits ..^
13	10, 12	eqtrd 2656	. . . . . . . . . . . . 13 bits bits ..^
14		bitsf1o 15167	. . . . . . . . . . . . . 14 bits
15		f1ocnvfv 6534	. . . . . . . . . . . . . 14 bits $/\$ bits bits ..^ bits bits ..^
16	14, 8, 15	sylancr 695	. . . . . . . . . . . . 13 bits bits ..^ bits bits ..^
17	13, 16	mpd 15	. . . . . . . . . . . 12 bits bits ..^
18	2, 17	syl5eq 2668	. . . . . . . . . . 11 bits ..^
19	18	oveq2d 6666	. . . . . . . . . 10 bits ..^
20	19	oveq1d 6665	. . . . . . . . 9 bits ..^
21	3	zred 11482	. . . . . . . . . 10
22	7	nnrpd 11870	. . . . . . . . . 10
23		moddifz 12682	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 693	. . . . . . . . 9
25	20, 24	eqeltrd 2701	. . . . . . . 8 bits ..^
26	7	nnzd 11481	. . . . . . . . 9
27	7	nnne0d 11065	. . . . . . . . 9
28		inss1 3833	. . . . . . . . . . . . . 14 bits ..^ bits
29		bitsss 15148	. . . . . . . . . . . . . 14 bits
30	28, 29	sstri 3612	. . . . . . . . . . . . 13 bits ..^
31		fzofi 12773	. . . . . . . . . . . . . 14 ..^
32		inss2 3834	. . . . . . . . . . . . . 14 bits ..^ ..^
33		ssfi 8180	. . . . . . . . . . . . . 14 ..^ $/\$ bits ..^ ..^ bits ..^
34	31, 32, 33	mp2an 708	. . . . . . . . . . . . 13 bits ..^
35		elfpw 8268	. . . . . . . . . . . . 13 bits ..^ bits ..^ $/\$ bits ..^
36	30, 34, 35	mpbir2an 955	. . . . . . . . . . . 12 bits ..^
37		f1ocnv 6149	. . . . . . . . . . . . . . 15 bits bits
38		f1of 6137	. . . . . . . . . . . . . . 15 bits bits
39	14, 37, 38	mp2b 10	. . . . . . . . . . . . . 14 bits
40	1	feq1i 6036	. . . . . . . . . . . . . 14 bits
41	39, 40	mpbir 221	. . . . . . . . . . . . 13
42	41	ffvelrni 6358	. . . . . . . . . . . 12 bits ..^ bits ..^
43	36, 42	mp1i 13	. . . . . . . . . . 11 bits ..^
44	43	nn0zd 11480	. . . . . . . . . 10 bits ..^
45	3, 44	zsubcld 11487	. . . . . . . . 9 bits ..^
46		dvdsval2 14986	. . . . . . . . 9 $/\$ $/\$ bits ..^ bits ..^ bits ..^
47	26, 27, 45, 46	syl3anc 1326	. . . . . . . 8 bits ..^ bits ..^
48	25, 47	mpbird 247	. . . . . . 7 bits ..^
49	1	fveq1i 6192	. . . . . . . . . . . 12 bits ..^ bits bits ..^
50		sadaddlem.2	. . . . . . . . . . . . . . . 16
51	50, 7	zmodcld 12691	. . . . . . . . . . . . . . 15
52		fvres 6207	. . . . . . . . . . . . . . 15 bits bits
53	51, 52	syl 17	. . . . . . . . . . . . . 14 bits bits
54		bitsmod 15158	. . . . . . . . . . . . . . 15 $/\$ bits bits ..^
55	50, 6, 54	syl2anc 693	. . . . . . . . . . . . . 14 bits bits ..^
56	53, 55	eqtrd 2656	. . . . . . . . . . . . 13 bits bits ..^
57		f1ocnvfv 6534	. . . . . . . . . . . . . 14 bits $/\$ bits bits ..^ bits bits ..^
58	14, 51, 57	sylancr 695	. . . . . . . . . . . . 13 bits bits ..^ bits bits ..^
59	56, 58	mpd 15	. . . . . . . . . . . 12 bits bits ..^
60	49, 59	syl5eq 2668	. . . . . . . . . . 11 bits ..^
61	60	oveq2d 6666	. . . . . . . . . 10 bits ..^
62	61	oveq1d 6665	. . . . . . . . 9 bits ..^
63	50	zred 11482	. . . . . . . . . 10
64		moddifz 12682	. . . . . . . . . 10 $/\$
65	63, 22, 64	syl2anc 693	. . . . . . . . 9
66	62, 65	eqeltrd 2701	. . . . . . . 8 bits ..^
67		inss1 3833	. . . . . . . . . . . . . 14 bits ..^ bits
68		bitsss 15148	. . . . . . . . . . . . . 14 bits
69	67, 68	sstri 3612	. . . . . . . . . . . . 13 bits ..^
70		inss2 3834	. . . . . . . . . . . . . 14 bits ..^ ..^
71		ssfi 8180	. . . . . . . . . . . . . 14 ..^ $/\$ bits ..^ ..^ bits ..^
72	31, 70, 71	mp2an 708	. . . . . . . . . . . . 13 bits ..^
73		elfpw 8268	. . . . . . . . . . . . 13 bits ..^ bits ..^ $/\$ bits ..^
74	69, 72, 73	mpbir2an 955	. . . . . . . . . . . 12 bits ..^
75	41	ffvelrni 6358	. . . . . . . . . . . 12 bits ..^ bits ..^
76	74, 75	mp1i 13	. . . . . . . . . . 11 bits ..^
77	76	nn0zd 11480	. . . . . . . . . 10 bits ..^
78	50, 77	zsubcld 11487	. . . . . . . . 9 bits ..^
79		dvdsval2 14986	. . . . . . . . 9 $/\$ $/\$ bits ..^ bits ..^ bits ..^
80	26, 27, 78, 79	syl3anc 1326	. . . . . . . 8 bits ..^ bits ..^
81	66, 80	mpbird 247	. . . . . . 7 bits ..^
82		dvds2add 15015	. . . . . . . 8 $/\$ bits ..^ $/\$ bits ..^ bits ..^ $/\$ bits ..^ bits ..^ bits ..^
83	26, 45, 78, 82	syl3anc 1326	. . . . . . 7 bits ..^ $/\$ bits ..^ bits ..^ bits ..^
84	48, 81, 83	mp2and 715	. . . . . 6 bits ..^ bits ..^
85	3	zcnd 11483	. . . . . . 7
86	50	zcnd 11483	. . . . . . 7
87	43	nn0cnd 11353	. . . . . . 7 bits ..^
88	76	nn0cnd 11353	. . . . . . 7 bits ..^
89	85, 86, 87, 88	addsub4d 10439	. . . . . 6 bits ..^ bits ..^ bits ..^ bits ..^
90	84, 89	breqtrrd 4681	. . . . 5 bits ..^ bits ..^
91	3, 50	zaddcld 11486	. . . . . 6
92	44, 77	zaddcld 11486	. . . . . 6 bits ..^ bits ..^
93		moddvds 14991	. . . . . 6 $/\$ $/\$ bits ..^ bits ..^ bits ..^ bits ..^ bits ..^ bits ..^
94	7, 91, 92, 93	syl3anc 1326	. . . . 5 bits ..^ bits ..^ bits ..^ bits ..^
95	90, 94	mpbird 247	. . . 4 bits ..^ bits ..^
96	29	a1i 11	. . . . 5 bits
97	68	a1i 11	. . . . 5 bits
98		sadaddlem.c	. . . . 5 cadd bits bits
99	96, 97, 98, 6, 1	sadadd3 15183	. . . 4 bits sadd bits ..^ bits ..^ bits ..^
100		inss1 3833	. . . . . . . . 9 bits sadd bits ..^ bits sadd bits
101		sadcl 15184	. . . . . . . . . 10 bits $/\$ bits bits sadd bits
102	29, 68, 101	mp2an 708	. . . . . . . . 9 bits sadd bits
103	100, 102	sstri 3612	. . . . . . . 8 bits sadd bits ..^
104		inss2 3834	. . . . . . . . 9 bits sadd bits ..^ ..^
105		ssfi 8180	. . . . . . . . 9 ..^ $/\$ bits sadd bits ..^ ..^ bits sadd bits ..^
106	31, 104, 105	mp2an 708	. . . . . . . 8 bits sadd bits ..^
107		elfpw 8268	. . . . . . . 8 bits sadd bits ..^ bits sadd bits ..^ $/\$ bits sadd bits ..^
108	103, 106, 107	mpbir2an 955	. . . . . . 7 bits sadd bits ..^
109	41	ffvelrni 6358	. . . . . . 7 bits sadd bits ..^ bits sadd bits ..^
110	108, 109	mp1i 13	. . . . . 6 bits sadd bits ..^
111	110	nn0red 11352	. . . . 5 bits sadd bits ..^
112	110	nn0ge0d 11354	. . . . 5 bits sadd bits ..^
113	1	fveq1i 6192	. . . . . . . . . 10 bits sadd bits ..^ bits bits sadd bits ..^
114	113	fveq2i 6194	. . . . . . . . 9 bits bits sadd bits ..^ bits bits bits sadd bits ..^
115		fvres 6207	. . . . . . . . . 10 bits sadd bits ..^ bits bits sadd bits ..^ bitsbits sadd bits ..^
116	110, 115	syl 17	. . . . . . . . 9 bits bits sadd bits ..^ bitsbits sadd bits ..^
117	108	a1i 11	. . . . . . . . . 10 bits sadd bits ..^
118		f1ocnvfv2 6533	. . . . . . . . . 10 bits $/\$ bits sadd bits ..^ bits bits bits sadd bits ..^ bits sadd bits ..^
119	14, 117, 118	sylancr 695	. . . . . . . . 9 bits bits bits sadd bits ..^ bits sadd bits ..^
120	114, 116, 119	3eqtr3a 2680	. . . . . . . 8 bitsbits sadd bits ..^ bits sadd bits ..^
121	120, 104	syl6eqss 3655	. . . . . . 7 bitsbits sadd bits ..^ ..^
122	110	nn0zd 11480	. . . . . . . 8 bits sadd bits ..^
123		bitsfzo 15157	. . . . . . . 8 bits sadd bits ..^ $/\$ bits sadd bits ..^ ..^ bitsbits sadd bits ..^ ..^
124	122, 6, 123	syl2anc 693	. . . . . . 7 bits sadd bits ..^ ..^ bitsbits sadd bits ..^ ..^
125	121, 124	mpbird 247	. . . . . 6 bits sadd bits ..^ ..^
126		elfzolt2 12479	. . . . . 6 bits sadd bits ..^ ..^ bits sadd bits ..^
127	125, 126	syl 17	. . . . 5 bits sadd bits ..^
128		modid 12695	. . . . 5 bits sadd bits ..^ $/\$ $/\$ bits sadd bits ..^ $/\$ bits sadd bits ..^ bits sadd bits ..^ bits sadd bits ..^
129	111, 22, 112, 127, 128	syl22anc 1327	. . . 4 bits sadd bits ..^ bits sadd bits ..^
130	95, 99, 129	3eqtr2d 2662	. . 3 bits sadd bits ..^
131	130	fveq2d 6195	. 2 bits bitsbits sadd bits ..^
132	131, 120	eqtr2d 2657	1 bits sadd bits ..^ bits

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483 caddwcad 1545

wcel 1990

wne 2794

cin 3573

wss 3574

c0 3915

cif 4086

cpw 4158 class class class wbr 4653

cmpt 4729

ccnv 5113

cres 5116

wf 5884

wf1o 5887

cfv 5888 (class class class)co 6650

cmpt2 6652

c1o 7553

c2o 7554

cfn 7955

cr 9935

cc0 9936

c1 9937

caddc 9939

clt 10074

cle 10075

cmin 10266

cdiv 10684

cn 11020

c2 11070

cn0 11292

cz 11377

crp 11832 ..^cfzo 12465

cmo 12668

cseq 12801

cexp 12860

cdvds 14983 bitscbits 15141 sadd csad 15142

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

This theorem is referenced by: sadadd 15189

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