sadaddlem - Metamath Proof Explorer

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Theorem sadaddlem 14440

Description: Lemma for sadadd 14441. (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 5866	. . . . . . . . . . . 12 bits ..^ bits bits ..^
3		sadaddlem.1	. . . . . . . . . . . . . . . 16
4		2nn 10767	. . . . . . . . . . . . . . . . . 18
5	4	a1i 11	. . . . . . . . . . . . . . . . 17
6		sadaddlem.3	. . . . . . . . . . . . . . . . 17
7	5, 6	nnexpcld 12437	. . . . . . . . . . . . . . . 16
8	3, 7	zmodcld 12117	. . . . . . . . . . . . . . 15
9		fvres 5879	. . . . . . . . . . . . . . 15 bits bits
10	8, 9	syl 17	. . . . . . . . . . . . . 14 bits bits
11		bitsmod 14410	. . . . . . . . . . . . . . 15 $/\$ bits bits ..^
12	3, 6, 11	syl2anc 667	. . . . . . . . . . . . . 14 bits bits ..^
13	10, 12	eqtrd 2485	. . . . . . . . . . . . 13 bits bits ..^
14		bitsf1o 14419	. . . . . . . . . . . . . 14 bits
15		f1ocnvfv 6177	. . . . . . . . . . . . . 14 bits $/\$ bits bits ..^ bits bits ..^
16	14, 8, 15	sylancr 669	. . . . . . . . . . . . 13 bits bits ..^ bits bits ..^
17	13, 16	mpd 15	. . . . . . . . . . . 12 bits bits ..^
18	2, 17	syl5eq 2497	. . . . . . . . . . 11 bits ..^
19	18	oveq2d 6306	. . . . . . . . . 10 bits ..^
20	19	oveq1d 6305	. . . . . . . . 9 bits ..^
21	3	zred 11040	. . . . . . . . . 10
22	7	nnrpd 11339	. . . . . . . . . 10
23		moddifz 12109	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 667	. . . . . . . . 9
25	20, 24	eqeltrd 2529	. . . . . . . 8 bits ..^
26	7	nnzd 11039	. . . . . . . . 9
27	7	nnne0d 10654	. . . . . . . . 9
28		inss1 3652	. . . . . . . . . . . . . 14 bits ..^ bits
29		bitsss 14399	. . . . . . . . . . . . . 14 bits
30	28, 29	sstri 3441	. . . . . . . . . . . . 13 bits ..^
31		fzofi 12187	. . . . . . . . . . . . . 14 ..^
32		inss2 3653	. . . . . . . . . . . . . 14 bits ..^ ..^
33		ssfi 7792	. . . . . . . . . . . . . 14 ..^ $/\$ bits ..^ ..^ bits ..^
34	31, 32, 33	mp2an 678	. . . . . . . . . . . . 13 bits ..^
35		elfpw 7876	. . . . . . . . . . . . 13 bits ..^ bits ..^ $/\$ bits ..^
36	30, 34, 35	mpbir2an 931	. . . . . . . . . . . 12 bits ..^
37		f1ocnv 5826	. . . . . . . . . . . . . . 15 bits bits
38		f1of 5814	. . . . . . . . . . . . . . 15 bits bits
39	14, 37, 38	mp2b 10	. . . . . . . . . . . . . 14 bits
40	1	feq1i 5720	. . . . . . . . . . . . . 14 bits
41	39, 40	mpbir 213	. . . . . . . . . . . . 13
42	41	ffvelrni 6021	. . . . . . . . . . . 12 bits ..^ bits ..^
43	36, 42	mp1i 13	. . . . . . . . . . 11 bits ..^
44	43	nn0zd 11038	. . . . . . . . . 10 bits ..^
45	3, 44	zsubcld 11045	. . . . . . . . 9 bits ..^
46		dvdsval2 14308	. . . . . . . . 9 $/\$ $/\$ bits ..^ bits ..^ bits ..^
47	26, 27, 45, 46	syl3anc 1268	. . . . . . . 8 bits ..^ bits ..^
48	25, 47	mpbird 236	. . . . . . 7 bits ..^
49	1	fveq1i 5866	. . . . . . . . . . . 12 bits ..^ bits bits ..^
50		sadaddlem.2	. . . . . . . . . . . . . . . 16
51	50, 7	zmodcld 12117	. . . . . . . . . . . . . . 15
52		fvres 5879	. . . . . . . . . . . . . . 15 bits bits
53	51, 52	syl 17	. . . . . . . . . . . . . 14 bits bits
54		bitsmod 14410	. . . . . . . . . . . . . . 15 $/\$ bits bits ..^
55	50, 6, 54	syl2anc 667	. . . . . . . . . . . . . 14 bits bits ..^
56	53, 55	eqtrd 2485	. . . . . . . . . . . . 13 bits bits ..^
57		f1ocnvfv 6177	. . . . . . . . . . . . . 14 bits $/\$ bits bits ..^ bits bits ..^
58	14, 51, 57	sylancr 669	. . . . . . . . . . . . 13 bits bits ..^ bits bits ..^
59	56, 58	mpd 15	. . . . . . . . . . . 12 bits bits ..^
60	49, 59	syl5eq 2497	. . . . . . . . . . 11 bits ..^
61	60	oveq2d 6306	. . . . . . . . . 10 bits ..^
62	61	oveq1d 6305	. . . . . . . . 9 bits ..^
63	50	zred 11040	. . . . . . . . . 10
64		moddifz 12109	. . . . . . . . . 10 $/\$
65	63, 22, 64	syl2anc 667	. . . . . . . . 9
66	62, 65	eqeltrd 2529	. . . . . . . 8 bits ..^
67		inss1 3652	. . . . . . . . . . . . . 14 bits ..^ bits
68		bitsss 14399	. . . . . . . . . . . . . 14 bits
69	67, 68	sstri 3441	. . . . . . . . . . . . 13 bits ..^
70		inss2 3653	. . . . . . . . . . . . . 14 bits ..^ ..^
71		ssfi 7792	. . . . . . . . . . . . . 14 ..^ $/\$ bits ..^ ..^ bits ..^
72	31, 70, 71	mp2an 678	. . . . . . . . . . . . 13 bits ..^
73		elfpw 7876	. . . . . . . . . . . . 13 bits ..^ bits ..^ $/\$ bits ..^
74	69, 72, 73	mpbir2an 931	. . . . . . . . . . . 12 bits ..^
75	41	ffvelrni 6021	. . . . . . . . . . . 12 bits ..^ bits ..^
76	74, 75	mp1i 13	. . . . . . . . . . 11 bits ..^
77	76	nn0zd 11038	. . . . . . . . . 10 bits ..^
78	50, 77	zsubcld 11045	. . . . . . . . 9 bits ..^
79		dvdsval2 14308	. . . . . . . . 9 $/\$ $/\$ bits ..^ bits ..^ bits ..^
80	26, 27, 78, 79	syl3anc 1268	. . . . . . . 8 bits ..^ bits ..^
81	66, 80	mpbird 236	. . . . . . 7 bits ..^
82		dvds2add 14334	. . . . . . . 8 $/\$ bits ..^ $/\$ bits ..^ bits ..^ $/\$ bits ..^ bits ..^ bits ..^
83	26, 45, 78, 82	syl3anc 1268	. . . . . . 7 bits ..^ $/\$ bits ..^ bits ..^ bits ..^
84	48, 81, 83	mp2and 685	. . . . . 6 bits ..^ bits ..^
85	3	zcnd 11041	. . . . . . 7
86	50	zcnd 11041	. . . . . . 7
87	43	nn0cnd 10927	. . . . . . 7 bits ..^
88	76	nn0cnd 10927	. . . . . . 7 bits ..^
89	85, 86, 87, 88	addsub4d 10033	. . . . . 6 bits ..^ bits ..^ bits ..^ bits ..^
90	84, 89	breqtrrd 4429	. . . . 5 bits ..^ bits ..^
91	3, 50	zaddcld 11044	. . . . . 6
92	44, 77	zaddcld 11044	. . . . . 6 bits ..^ bits ..^
93		moddvds 14312	. . . . . 6 $/\$ $/\$ bits ..^ bits ..^ bits ..^ bits ..^ bits ..^ bits ..^
94	7, 91, 92, 93	syl3anc 1268	. . . . 5 bits ..^ bits ..^ bits ..^ bits ..^
95	90, 94	mpbird 236	. . . 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 14435	. . . 4 bits sadd bits ..^ bits ..^ bits ..^
100		inss1 3652	. . . . . . . . 9 bits sadd bits ..^ bits sadd bits
101		sadcl 14436	. . . . . . . . . 10 bits $/\$ bits bits sadd bits
102	29, 68, 101	mp2an 678	. . . . . . . . 9 bits sadd bits
103	100, 102	sstri 3441	. . . . . . . 8 bits sadd bits ..^
104		inss2 3653	. . . . . . . . 9 bits sadd bits ..^ ..^
105		ssfi 7792	. . . . . . . . 9 ..^ $/\$ bits sadd bits ..^ ..^ bits sadd bits ..^
106	31, 104, 105	mp2an 678	. . . . . . . 8 bits sadd bits ..^
107		elfpw 7876	. . . . . . . 8 bits sadd bits ..^ bits sadd bits ..^ $/\$ bits sadd bits ..^
108	103, 106, 107	mpbir2an 931	. . . . . . 7 bits sadd bits ..^
109	41	ffvelrni 6021	. . . . . . 7 bits sadd bits ..^ bits sadd bits ..^
110	108, 109	mp1i 13	. . . . . 6 bits sadd bits ..^
111	110	nn0red 10926	. . . . 5 bits sadd bits ..^
112	110	nn0ge0d 10928	. . . . 5 bits sadd bits ..^
113	1	fveq1i 5866	. . . . . . . . . 10 bits sadd bits ..^ bits bits sadd bits ..^
114	113	fveq2i 5868	. . . . . . . . 9 bits bits sadd bits ..^ bits bits bits sadd bits ..^
115		fvres 5879	. . . . . . . . . 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 6176	. . . . . . . . . 10 bits $/\$ bits sadd bits ..^ bits bits bits sadd bits ..^ bits sadd bits ..^
119	14, 117, 118	sylancr 669	. . . . . . . . 9 bits bits bits sadd bits ..^ bits sadd bits ..^
120	114, 116, 119	3eqtr3a 2509	. . . . . . . 8 bitsbits sadd bits ..^ bits sadd bits ..^
121	120, 104	syl6eqss 3482	. . . . . . 7 bitsbits sadd bits ..^ ..^
122	110	nn0zd 11038	. . . . . . . 8 bits sadd bits ..^
123		bitsfzo 14409	. . . . . . . 8 bits sadd bits ..^ $/\$ bits sadd bits ..^ ..^ bitsbits sadd bits ..^ ..^
124	122, 6, 123	syl2anc 667	. . . . . . 7 bits sadd bits ..^ ..^ bitsbits sadd bits ..^ ..^
125	121, 124	mpbird 236	. . . . . 6 bits sadd bits ..^ ..^
126		elfzolt2 11929	. . . . . 6 bits sadd bits ..^ ..^ bits sadd bits ..^
127	125, 126	syl 17	. . . . 5 bits sadd bits ..^
128		modid 12121	. . . . 5 bits sadd bits ..^ $/\$ $/\$ bits sadd bits ..^ $/\$ bits sadd bits ..^ bits sadd bits ..^ bits sadd bits ..^
129	111, 22, 112, 127, 128	syl22anc 1269	. . . 4 bits sadd bits ..^ bits sadd bits ..^
130	95, 99, 129	3eqtr2d 2491	. . 3 bits sadd bits ..^
131	130	fveq2d 5869	. 2 bits bitsbits sadd bits ..^
132	131, 120	eqtr2d 2486	1 bits sadd bits ..^ bits

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 188 $/\$ wa 371

wceq 1444 caddwcad 1509

wcel 1887

wne 2622

cin 3403

wss 3404

c0 3731

cif 3881

cpw 3951 class class class wbr 4402

cmpt 4461

ccnv 4833

cres 4836

wf 5578

wf1o 5581

cfv 5582 (class class class)co 6290

cmpt2 6292

c1o 7175

c2o 7176

cfn 7569

cr 9538

cc0 9539

c1 9540

caddc 9542

clt 9675

cle 9676

cmin 9860

cdiv 10269

cn 10609

c2 10659

cn0 10869

cz 10937

crp 11302 ..^cfzo 11915

cmo 12096

cseq 12213

cexp 12272

cdvds 14305 bitscbits 14392 sadd csad 14393

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-rep 4515 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-inf2 8146 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 ax-pre-sup 9617

This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-xor 1406 df-tru 1447 df-fal 1450 df-had 1497 df-cad 1510 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-pss 3420 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-tp 3973 df-op 3975 df-uni 4199 df-int 4235 df-iun 4280 df-disj 4374 df-br 4403 df-opab 4462 df-mpt 4463 df-tr 4498 df-eprel 4745 df-id 4749 df-po 4755 df-so 4756 df-fr 4793 df-se 4794 df-we 4795 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-pred 5380 df-ord 5426 df-on 5427 df-lim 5428 df-suc 5429 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-isom 5591 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-om 6693 df-1st 6793 df-2nd 6794 df-wrecs 7028 df-recs 7090 df-rdg 7128 df-1o 7182 df-2o 7183 df-oadd 7186 df-er 7363 df-map 7474 df-pm 7475 df-en 7570 df-dom 7571 df-sdom 7572 df-fin 7573 df-sup 7956 df-inf 7957 df-oi 8025 df-card 8373 df-cda 8598 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 df-div 10270 df-nn 10610 df-2 10668 df-3 10669 df-n0 10870 df-z 10938 df-uz 11160 df-rp 11303 df-fz 11785 df-fzo 11916 df-fl 12028 df-mod 12097 df-seq 12214 df-exp 12273 df-hash 12516 df-cj 13162 df-re 13163 df-im 13164 df-sqrt 13298 df-abs 13299 df-clim 13552 df-sum 13753 df-dvds 14306 df-bits 14395 df-sad 14425

This theorem is referenced by: sadadd 14441

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