sadaddlem - Metamath Proof Explorer

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Theorem sadaddlem 13683

Description: Lemma for sadadd 13684. (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 5713	. . . . . . . . . . . 12 bits ..^ bits bits ..^
3		sadaddlem.1	. . . . . . . . . . . . . . . 16
4		2nn 10500	. . . . . . . . . . . . . . . . . 18
5	4	a1i 11	. . . . . . . . . . . . . . . . 17
6		sadaddlem.3	. . . . . . . . . . . . . . . . 17
7	5, 6	nnexpcld 12050	. . . . . . . . . . . . . . . 16
8	3, 7	zmodcld 11749	. . . . . . . . . . . . . . 15
9		fvres 5725	. . . . . . . . . . . . . . 15 bits bits
10	8, 9	syl 16	. . . . . . . . . . . . . 14 bits bits
11		bitsmod 13653	. . . . . . . . . . . . . . 15 $/\$ bits bits ..^
12	3, 6, 11	syl2anc 661	. . . . . . . . . . . . . 14 bits bits ..^
13	10, 12	eqtrd 2475	. . . . . . . . . . . . 13 bits bits ..^
14		bitsf1o 13662	. . . . . . . . . . . . . 14 bits
15		f1ocnvfv 6006	. . . . . . . . . . . . . 14 bits $/\$ bits bits ..^ bits bits ..^
16	14, 8, 15	sylancr 663	. . . . . . . . . . . . 13 bits bits ..^ bits bits ..^
17	13, 16	mpd 15	. . . . . . . . . . . 12 bits bits ..^
18	2, 17	syl5eq 2487	. . . . . . . . . . 11 bits ..^
19	18	oveq2d 6128	. . . . . . . . . 10 bits ..^
20	19	oveq1d 6127	. . . . . . . . 9 bits ..^
21	3	zred 10768	. . . . . . . . . 10
22	7	nnrpd 11047	. . . . . . . . . 10
23		moddifz 11741	. . . . . . . . . 10 $/\$
24	21, 22, 23	syl2anc 661	. . . . . . . . 9
25	20, 24	eqeltrd 2517	. . . . . . . 8 bits ..^
26	7	nnzd 10767	. . . . . . . . 9
27	7	nnne0d 10387	. . . . . . . . 9
28		inss1 3591	. . . . . . . . . . . . . 14 bits ..^ bits
29		bitsss 13643	. . . . . . . . . . . . . 14 bits
30	28, 29	sstri 3386	. . . . . . . . . . . . 13 bits ..^
31		fzofi 11817	. . . . . . . . . . . . . 14 ..^
32		inss2 3592	. . . . . . . . . . . . . 14 bits ..^ ..^
33		ssfi 7554	. . . . . . . . . . . . . 14 ..^ $/\$ bits ..^ ..^ bits ..^
34	31, 32, 33	mp2an 672	. . . . . . . . . . . . 13 bits ..^
35		elfpw 7634	. . . . . . . . . . . . 13 bits ..^ bits ..^ $/\$ bits ..^
36	30, 34, 35	mpbir2an 911	. . . . . . . . . . . 12 bits ..^
37		f1ocnv 5674	. . . . . . . . . . . . . . 15 bits bits
38		f1of 5662	. . . . . . . . . . . . . . 15 bits bits
39	14, 37, 38	mp2b 10	. . . . . . . . . . . . . 14 bits
40	1	feq1i 5572	. . . . . . . . . . . . . 14 bits
41	39, 40	mpbir 209	. . . . . . . . . . . . 13
42	41	ffvelrni 5863	. . . . . . . . . . . 12 bits ..^ bits ..^
43	36, 42	mp1i 12	. . . . . . . . . . 11 bits ..^
44	43	nn0zd 10766	. . . . . . . . . 10 bits ..^
45	3, 44	zsubcld 10773	. . . . . . . . 9 bits ..^
46		dvdsval2 13559	. . . . . . . . 9 $/\$ $/\$ bits ..^ bits ..^ bits ..^
47	26, 27, 45, 46	syl3anc 1218	. . . . . . . 8 bits ..^ bits ..^
48	25, 47	mpbird 232	. . . . . . 7 bits ..^
49	1	fveq1i 5713	. . . . . . . . . . . 12 bits ..^ bits bits ..^
50		sadaddlem.2	. . . . . . . . . . . . . . . 16
51	50, 7	zmodcld 11749	. . . . . . . . . . . . . . 15
52		fvres 5725	. . . . . . . . . . . . . . 15 bits bits
53	51, 52	syl 16	. . . . . . . . . . . . . 14 bits bits
54		bitsmod 13653	. . . . . . . . . . . . . . 15 $/\$ bits bits ..^
55	50, 6, 54	syl2anc 661	. . . . . . . . . . . . . 14 bits bits ..^
56	53, 55	eqtrd 2475	. . . . . . . . . . . . 13 bits bits ..^
57		f1ocnvfv 6006	. . . . . . . . . . . . . 14 bits $/\$ bits bits ..^ bits bits ..^
58	14, 51, 57	sylancr 663	. . . . . . . . . . . . 13 bits bits ..^ bits bits ..^
59	56, 58	mpd 15	. . . . . . . . . . . 12 bits bits ..^
60	49, 59	syl5eq 2487	. . . . . . . . . . 11 bits ..^
61	60	oveq2d 6128	. . . . . . . . . 10 bits ..^
62	61	oveq1d 6127	. . . . . . . . 9 bits ..^
63	50	zred 10768	. . . . . . . . . 10
64		moddifz 11741	. . . . . . . . . 10 $/\$
65	63, 22, 64	syl2anc 661	. . . . . . . . 9
66	62, 65	eqeltrd 2517	. . . . . . . 8 bits ..^
67		inss1 3591	. . . . . . . . . . . . . 14 bits ..^ bits
68		bitsss 13643	. . . . . . . . . . . . . 14 bits
69	67, 68	sstri 3386	. . . . . . . . . . . . 13 bits ..^
70		inss2 3592	. . . . . . . . . . . . . 14 bits ..^ ..^
71		ssfi 7554	. . . . . . . . . . . . . 14 ..^ $/\$ bits ..^ ..^ bits ..^
72	31, 70, 71	mp2an 672	. . . . . . . . . . . . 13 bits ..^
73		elfpw 7634	. . . . . . . . . . . . 13 bits ..^ bits ..^ $/\$ bits ..^
74	69, 72, 73	mpbir2an 911	. . . . . . . . . . . 12 bits ..^
75	41	ffvelrni 5863	. . . . . . . . . . . 12 bits ..^ bits ..^
76	74, 75	mp1i 12	. . . . . . . . . . 11 bits ..^
77	76	nn0zd 10766	. . . . . . . . . 10 bits ..^
78	50, 77	zsubcld 10773	. . . . . . . . 9 bits ..^
79		dvdsval2 13559	. . . . . . . . 9 $/\$ $/\$ bits ..^ bits ..^ bits ..^
80	26, 27, 78, 79	syl3anc 1218	. . . . . . . 8 bits ..^ bits ..^
81	66, 80	mpbird 232	. . . . . . 7 bits ..^
82		dvds2add 13585	. . . . . . . 8 $/\$ bits ..^ $/\$ bits ..^ bits ..^ $/\$ bits ..^ bits ..^ bits ..^
83	26, 45, 78, 82	syl3anc 1218	. . . . . . 7 bits ..^ $/\$ bits ..^ bits ..^ bits ..^
84	48, 81, 83	mp2and 679	. . . . . 6 bits ..^ bits ..^
85	3	zcnd 10769	. . . . . . 7
86	50	zcnd 10769	. . . . . . 7
87	43	nn0cnd 10659	. . . . . . 7 bits ..^
88	76	nn0cnd 10659	. . . . . . 7 bits ..^
89	85, 86, 87, 88	addsub4d 9787	. . . . . 6 bits ..^ bits ..^ bits ..^ bits ..^
90	84, 89	breqtrrd 4339	. . . . 5 bits ..^ bits ..^
91	3, 50	zaddcld 10772	. . . . . 6
92	44, 77	zaddcld 10772	. . . . . 6 bits ..^ bits ..^
93		moddvds 13563	. . . . . 6 $/\$ $/\$ bits ..^ bits ..^ bits ..^ bits ..^ bits ..^ bits ..^
94	7, 91, 92, 93	syl3anc 1218	. . . . 5 bits ..^ bits ..^ bits ..^ bits ..^
95	90, 94	mpbird 232	. . . 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 13678	. . . 4 bits sadd bits ..^ bits ..^ bits ..^
100		inss1 3591	. . . . . . . . 9 bits sadd bits ..^ bits sadd bits
101		sadcl 13679	. . . . . . . . . 10 bits $/\$ bits bits sadd bits
102	29, 68, 101	mp2an 672	. . . . . . . . 9 bits sadd bits
103	100, 102	sstri 3386	. . . . . . . 8 bits sadd bits ..^
104		inss2 3592	. . . . . . . . 9 bits sadd bits ..^ ..^
105		ssfi 7554	. . . . . . . . 9 ..^ $/\$ bits sadd bits ..^ ..^ bits sadd bits ..^
106	31, 104, 105	mp2an 672	. . . . . . . 8 bits sadd bits ..^
107		elfpw 7634	. . . . . . . 8 bits sadd bits ..^ bits sadd bits ..^ $/\$ bits sadd bits ..^
108	103, 106, 107	mpbir2an 911	. . . . . . 7 bits sadd bits ..^
109	41	ffvelrni 5863	. . . . . . 7 bits sadd bits ..^ bits sadd bits ..^
110	108, 109	mp1i 12	. . . . . 6 bits sadd bits ..^
111	110	nn0red 10658	. . . . 5 bits sadd bits ..^
112	110	nn0ge0d 10660	. . . . 5 bits sadd bits ..^
113	1	fveq1i 5713	. . . . . . . . . 10 bits sadd bits ..^ bits bits sadd bits ..^
114	113	fveq2i 5715	. . . . . . . . 9 bits bits sadd bits ..^ bits bits bits sadd bits ..^
115		fvres 5725	. . . . . . . . . 10 bits sadd bits ..^ bits bits sadd bits ..^ bitsbits sadd bits ..^
116	110, 115	syl 16	. . . . . . . . 9 bits bits sadd bits ..^ bitsbits sadd bits ..^
117	108	a1i 11	. . . . . . . . . 10 bits sadd bits ..^
118		f1ocnvfv2 6005	. . . . . . . . . 10 bits $/\$ bits sadd bits ..^ bits bits bits sadd bits ..^ bits sadd bits ..^
119	14, 117, 118	sylancr 663	. . . . . . . . 9 bits bits bits sadd bits ..^ bits sadd bits ..^
120	114, 116, 119	3eqtr3a 2499	. . . . . . . 8 bitsbits sadd bits ..^ bits sadd bits ..^
121	120, 104	syl6eqss 3427	. . . . . . 7 bitsbits sadd bits ..^ ..^
122	110	nn0zd 10766	. . . . . . . 8 bits sadd bits ..^
123		bitsfzo 13652	. . . . . . . 8 bits sadd bits ..^ $/\$ bits sadd bits ..^ ..^ bitsbits sadd bits ..^ ..^
124	122, 6, 123	syl2anc 661	. . . . . . 7 bits sadd bits ..^ ..^ bitsbits sadd bits ..^ ..^
125	121, 124	mpbird 232	. . . . . 6 bits sadd bits ..^ ..^
126		elfzolt2 11582	. . . . . 6 bits sadd bits ..^ ..^ bits sadd bits ..^
127	125, 126	syl 16	. . . . 5 bits sadd bits ..^
128		modid 11753	. . . . 5 bits sadd bits ..^ $/\$ $/\$ bits sadd bits ..^ $/\$ bits sadd bits ..^ bits sadd bits ..^ bits sadd bits ..^
129	111, 22, 112, 127, 128	syl22anc 1219	. . . 4 bits sadd bits ..^ bits sadd bits ..^
130	95, 99, 129	3eqtr2d 2481	. . 3 bits sadd bits ..^
131	130	fveq2d 5716	. 2 bits bitsbits sadd bits ..^
132	131, 120	eqtr2d 2476	1 bits sadd bits ..^ bits

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 184 $/\$ wa 369

wceq 1369 caddwcad 1420

wcel 1756

wne 2620

cin 3348

wss 3349

c0 3658

cif 3812

cpw 3881 class class class wbr 4313

cmpt 4371

ccnv 4860

cres 4863

wf 5435

wf1o 5438

cfv 5439 (class class class)co 6112

cmpt2 6114

c1o 6934

c2o 6935

cfn 7331

cr 9302

cc0 9303

c1 9304

caddc 9306

clt 9439

cle 9440

cmin 9616

cdiv 10014

cn 10343

c2 10392

cn0 10600

cz 10667

crp 11012 ..^cfzo 11569

cmo 11729

cseq 11827

cexp 11886

cdivides 13556 bitscbits 13636 sadd csad 13637

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1591 ax-4 1602 ax-5 1670 ax-6 1708 ax-7 1728 ax-8 1758 ax-9 1760 ax-10 1775 ax-11 1780 ax-12 1792 ax-13 1943 ax-ext 2423 ax-rep 4424 ax-sep 4434 ax-nul 4442 ax-pow 4491 ax-pr 4552 ax-un 6393 ax-inf2 7868 ax-cnex 9359 ax-resscn 9360 ax-1cn 9361 ax-icn 9362 ax-addcl 9363 ax-addrcl 9364 ax-mulcl 9365 ax-mulrcl 9366 ax-mulcom 9367 ax-addass 9368 ax-mulass 9369 ax-distr 9370 ax-i2m1 9371 ax-1ne0 9372 ax-1rid 9373 ax-rnegex 9374 ax-rrecex 9375 ax-cnre 9376 ax-pre-lttri 9377 ax-pre-lttrn 9378 ax-pre-ltadd 9379 ax-pre-mulgt0 9380 ax-pre-sup 9381

This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-xor 1351 df-tru 1372 df-fal 1375 df-had 1421 df-cad 1422 df-ex 1587 df-nf 1590 df-sb 1701 df-eu 2257 df-mo 2258 df-clab 2430 df-cleq 2436 df-clel 2439 df-nfc 2577 df-ne 2622 df-nel 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 2995 df-sbc 3208 df-csb 3310 df-dif 3352 df-un 3354 df-in 3356 df-ss 3363 df-pss 3365 df-nul 3659 df-if 3813 df-pw 3883 df-sn 3899 df-pr 3901 df-tp 3903 df-op 3905 df-uni 4113 df-int 4150 df-iun 4194 df-disj 4284 df-br 4314 df-opab 4372 df-mpt 4373 df-tr 4407 df-eprel 4653 df-id 4657 df-po 4662 df-so 4663 df-fr 4700 df-se 4701 df-we 4702 df-ord 4743 df-on 4744 df-lim 4745 df-suc 4746 df-xp 4867 df-rel 4868 df-cnv 4869 df-co 4870 df-dm 4871 df-rn 4872 df-res 4873 df-ima 4874 df-iota 5402 df-fun 5441 df-fn 5442 df-f 5443 df-f1 5444 df-fo 5445 df-f1o 5446 df-fv 5447 df-isom 5448 df-riota 6073 df-ov 6115 df-oprab 6116 df-mpt2 6117 df-om 6498 df-1st 6598 df-2nd 6599 df-recs 6853 df-rdg 6887 df-1o 6941 df-2o 6942 df-oadd 6945 df-er 7122 df-map 7237 df-pm 7238 df-en 7332 df-dom 7333 df-sdom 7334 df-fin 7335 df-sup 7712 df-oi 7745 df-card 8130 df-cda 8358 df-pnf 9441 df-mnf 9442 df-xr 9443 df-ltxr 9444 df-le 9445 df-sub 9618 df-neg 9619 df-div 10015 df-nn 10344 df-2 10401 df-3 10402 df-n0 10601 df-z 10668 df-uz 10883 df-rp 11013 df-fz 11459 df-fzo 11570 df-fl 11663 df-mod 11730 df-seq 11828 df-exp 11887 df-hash 12125 df-cj 12609 df-re 12610 df-im 12611 df-sqr 12745 df-abs 12746 df-clim 12987 df-sum 13185 df-dvds 13557 df-bits 13639 df-sad 13668

This theorem is referenced by: sadadd 13684

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