sadasslem - Metamath Proof Explorer

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Theorem sadasslem 15192

Description: Lemma for sadass 15193. (Contributed by Mario Carneiro, 9-Sep-2016.)

**Hypotheses**
Ref	Expression
sadasslem.1
sadasslem.2
sadasslem.3
sadasslem.4

**Assertion**
Ref	Expression
sadasslem	sadd sadd ..^ sadd sadd ..^

Proof of Theorem sadasslem Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3833	. . . . . . . . . . 11 ..^
2		sadasslem.1	. . . . . . . . . . 11
3	1, 2	syl5ss 3614	. . . . . . . . . 10 ..^
4		fzofi 12773	. . . . . . . . . . . 12 ..^
5	4	a1i 11	. . . . . . . . . . 11 ..^
6		inss2 3834	. . . . . . . . . . 11 ..^ ..^
7		ssfi 8180	. . . . . . . . . . 11 ..^ $/\$ ..^ ..^ ..^
8	5, 6, 7	sylancl 694	. . . . . . . . . 10 ..^
9		elfpw 8268	. . . . . . . . . 10 ..^ ..^ $/\$ ..^
10	3, 8, 9	sylanbrc 698	. . . . . . . . 9 ..^
11		bitsf1o 15167	. . . . . . . . . . 11 bits
12		f1ocnv 6149	. . . . . . . . . . 11 bits bits
13		f1of 6137	. . . . . . . . . . 11 bits bits
14	11, 12, 13	mp2b 10	. . . . . . . . . 10 bits
15	14	ffvelrni 6358	. . . . . . . . 9 ..^ bits ..^
16	10, 15	syl 17	. . . . . . . 8 bits ..^
17	16	nn0cnd 11353	. . . . . . 7 bits ..^
18		inss1 3833	. . . . . . . . . . 11 ..^
19		sadasslem.2	. . . . . . . . . . 11
20	18, 19	syl5ss 3614	. . . . . . . . . 10 ..^
21		inss2 3834	. . . . . . . . . . 11 ..^ ..^
22		ssfi 8180	. . . . . . . . . . 11 ..^ $/\$ ..^ ..^ ..^
23	5, 21, 22	sylancl 694	. . . . . . . . . 10 ..^
24		elfpw 8268	. . . . . . . . . 10 ..^ ..^ $/\$ ..^
25	20, 23, 24	sylanbrc 698	. . . . . . . . 9 ..^
26	14	ffvelrni 6358	. . . . . . . . 9 ..^ bits ..^
27	25, 26	syl 17	. . . . . . . 8 bits ..^
28	27	nn0cnd 11353	. . . . . . 7 bits ..^
29		inss1 3833	. . . . . . . . . . 11 ..^
30		sadasslem.3	. . . . . . . . . . 11
31	29, 30	syl5ss 3614	. . . . . . . . . 10 ..^
32		inss2 3834	. . . . . . . . . . 11 ..^ ..^
33		ssfi 8180	. . . . . . . . . . 11 ..^ $/\$ ..^ ..^ ..^
34	5, 32, 33	sylancl 694	. . . . . . . . . 10 ..^
35		elfpw 8268	. . . . . . . . . 10 ..^ ..^ $/\$ ..^
36	31, 34, 35	sylanbrc 698	. . . . . . . . 9 ..^
37	14	ffvelrni 6358	. . . . . . . . 9 ..^ bits ..^
38	36, 37	syl 17	. . . . . . . 8 bits ..^
39	38	nn0cnd 11353	. . . . . . 7 bits ..^
40	17, 28, 39	addassd 10062	. . . . . 6 bits ..^ bits ..^ bits ..^ bits ..^ bits ..^ bits ..^
41	40	oveq1d 6665	. . . . 5 bits ..^ bits ..^ bits ..^ bits ..^ bits ..^ bits ..^
42		inss1 3833	. . . . . . . . . 10 sadd ..^ sadd
43		sadcl 15184	. . . . . . . . . . 11 $/\$ sadd
44	2, 19, 43	syl2anc 693	. . . . . . . . . 10 sadd
45	42, 44	syl5ss 3614	. . . . . . . . 9 sadd ..^
46		inss2 3834	. . . . . . . . . 10 sadd ..^ ..^
47		ssfi 8180	. . . . . . . . . 10 ..^ $/\$ sadd ..^ ..^ sadd ..^
48	5, 46, 47	sylancl 694	. . . . . . . . 9 sadd ..^
49		elfpw 8268	. . . . . . . . 9 sadd ..^ sadd ..^ $/\$ sadd ..^
50	45, 48, 49	sylanbrc 698	. . . . . . . 8 sadd ..^
51	14	ffvelrni 6358	. . . . . . . 8 sadd ..^ bits sadd ..^
52	50, 51	syl 17	. . . . . . 7 bits sadd ..^
53	52	nn0red 11352	. . . . . 6 bits sadd ..^
54	16	nn0red 11352	. . . . . . 7 bits ..^
55	27	nn0red 11352	. . . . . . 7 bits ..^
56	54, 55	readdcld 10069	. . . . . 6 bits ..^ bits ..^
57	38	nn0red 11352	. . . . . 6 bits ..^
58		2rp 11837	. . . . . . . 8
59	58	a1i 11	. . . . . . 7
60		sadasslem.4	. . . . . . . 8
61	60	nn0zd 11480	. . . . . . 7
62	59, 61	rpexpcld 13032	. . . . . 6
63		eqid 2622	. . . . . . 7 cadd cadd
64		eqid 2622	. . . . . . 7 bits bits
65	2, 19, 63, 60, 64	sadadd3 15183	. . . . . 6 bits sadd ..^ bits ..^ bits ..^
66		eqidd 2623	. . . . . 6 bits ..^ bits ..^
67	53, 56, 57, 57, 62, 65, 66	modadd12d 12726	. . . . 5 bits sadd ..^ bits ..^ bits ..^ bits ..^ bits ..^
68		inss1 3833	. . . . . . . . . 10 sadd ..^ sadd
69		sadcl 15184	. . . . . . . . . . 11 $/\$ sadd
70	19, 30, 69	syl2anc 693	. . . . . . . . . 10 sadd
71	68, 70	syl5ss 3614	. . . . . . . . 9 sadd ..^
72		inss2 3834	. . . . . . . . . 10 sadd ..^ ..^
73		ssfi 8180	. . . . . . . . . 10 ..^ $/\$ sadd ..^ ..^ sadd ..^
74	5, 72, 73	sylancl 694	. . . . . . . . 9 sadd ..^
75		elfpw 8268	. . . . . . . . 9 sadd ..^ sadd ..^ $/\$ sadd ..^
76	71, 74, 75	sylanbrc 698	. . . . . . . 8 sadd ..^
77	14	ffvelrni 6358	. . . . . . . 8 sadd ..^ bits sadd ..^
78	76, 77	syl 17	. . . . . . 7 bits sadd ..^
79	78	nn0red 11352	. . . . . 6 bits sadd ..^
80	55, 57	readdcld 10069	. . . . . 6 bits ..^ bits ..^
81		eqidd 2623	. . . . . 6 bits ..^ bits ..^
82		eqid 2622	. . . . . . 7 cadd cadd
83	19, 30, 82, 60, 64	sadadd3 15183	. . . . . 6 bits sadd ..^ bits ..^ bits ..^
84	54, 54, 79, 80, 62, 81, 83	modadd12d 12726	. . . . 5 bits ..^ bits sadd ..^ bits ..^ bits ..^ bits ..^
85	41, 67, 84	3eqtr4d 2666	. . . 4 bits sadd ..^ bits ..^ bits ..^ bits sadd ..^
86		eqid 2622	. . . . 5 cadd sadd cadd sadd
87	44, 30, 86, 60, 64	sadadd3 15183	. . . 4 bits sadd sadd ..^ bits sadd ..^ bits ..^
88		eqid 2622	. . . . 5 cadd sadd cadd sadd
89	2, 70, 88, 60, 64	sadadd3 15183	. . . 4 bits sadd sadd ..^ bits ..^ bits sadd ..^
90	85, 87, 89	3eqtr4d 2666	. . 3 bits sadd sadd ..^ bits sadd sadd ..^
91		inss1 3833	. . . . . . . 8 sadd sadd ..^ sadd sadd
92		sadcl 15184	. . . . . . . . 9 sadd $/\$ sadd sadd
93	44, 30, 92	syl2anc 693	. . . . . . . 8 sadd sadd
94	91, 93	syl5ss 3614	. . . . . . 7 sadd sadd ..^
95		inss2 3834	. . . . . . . 8 sadd sadd ..^ ..^
96		ssfi 8180	. . . . . . . 8 ..^ $/\$ sadd sadd ..^ ..^ sadd sadd ..^
97	5, 95, 96	sylancl 694	. . . . . . 7 sadd sadd ..^
98		elfpw 8268	. . . . . . 7 sadd sadd ..^ sadd sadd ..^ $/\$ sadd sadd ..^
99	94, 97, 98	sylanbrc 698	. . . . . 6 sadd sadd ..^
100	14	ffvelrni 6358	. . . . . 6 sadd sadd ..^ bits sadd sadd ..^
101	99, 100	syl 17	. . . . 5 bits sadd sadd ..^
102	101	nn0red 11352	. . . 4 bits sadd sadd ..^
103	101	nn0ge0d 11354	. . . 4 bits sadd sadd ..^
104		fvres 6207	. . . . . . . . 9 bits sadd sadd ..^ bits bits sadd sadd ..^ bitsbits sadd sadd ..^
105	101, 104	syl 17	. . . . . . . 8 bits bits sadd sadd ..^ bitsbits sadd sadd ..^
106		f1ocnvfv2 6533	. . . . . . . . 9 bits $/\$ sadd sadd ..^ bits bits sadd sadd ..^ sadd sadd ..^
107	11, 99, 106	sylancr 695	. . . . . . . 8 bits bits sadd sadd ..^ sadd sadd ..^
108	105, 107	eqtr3d 2658	. . . . . . 7 bitsbits sadd sadd ..^ sadd sadd ..^
109	108, 95	syl6eqss 3655	. . . . . 6 bitsbits sadd sadd ..^ ..^
110	101	nn0zd 11480	. . . . . . 7 bits sadd sadd ..^
111		bitsfzo 15157	. . . . . . 7 bits sadd sadd ..^ $/\$ bits sadd sadd ..^ ..^ bitsbits sadd sadd ..^ ..^
112	110, 60, 111	syl2anc 693	. . . . . 6 bits sadd sadd ..^ ..^ bitsbits sadd sadd ..^ ..^
113	109, 112	mpbird 247	. . . . 5 bits sadd sadd ..^ ..^
114		elfzolt2 12479	. . . . 5 bits sadd sadd ..^ ..^ bits sadd sadd ..^
115	113, 114	syl 17	. . . 4 bits sadd sadd ..^
116		modid 12695	. . . 4 bits sadd sadd ..^ $/\$ $/\$ bits sadd sadd ..^ $/\$ bits sadd sadd ..^ bits sadd sadd ..^ bits sadd sadd ..^
117	102, 62, 103, 115, 116	syl22anc 1327	. . 3 bits sadd sadd ..^ bits sadd sadd ..^
118		inss1 3833	. . . . . . . 8 sadd sadd ..^ sadd sadd
119		sadcl 15184	. . . . . . . . 9 $/\$ sadd sadd sadd
120	2, 70, 119	syl2anc 693	. . . . . . . 8 sadd sadd
121	118, 120	syl5ss 3614	. . . . . . 7 sadd sadd ..^
122		inss2 3834	. . . . . . . 8 sadd sadd ..^ ..^
123		ssfi 8180	. . . . . . . 8 ..^ $/\$ sadd sadd ..^ ..^ sadd sadd ..^
124	5, 122, 123	sylancl 694	. . . . . . 7 sadd sadd ..^
125		elfpw 8268	. . . . . . 7 sadd sadd ..^ sadd sadd ..^ $/\$ sadd sadd ..^
126	121, 124, 125	sylanbrc 698	. . . . . 6 sadd sadd ..^
127	14	ffvelrni 6358	. . . . . 6 sadd sadd ..^ bits sadd sadd ..^
128	126, 127	syl 17	. . . . 5 bits sadd sadd ..^
129	128	nn0red 11352	. . . 4 bits sadd sadd ..^
130		2nn 11185	. . . . . . 7
131	130	a1i 11	. . . . . 6
132	131, 60	nnexpcld 13030	. . . . 5
133	132	nnrpd 11870	. . . 4
134	128	nn0ge0d 11354	. . . 4 bits sadd sadd ..^
135		fvres 6207	. . . . . . . . 9 bits sadd sadd ..^ bits bits sadd sadd ..^ bitsbits sadd sadd ..^
136	128, 135	syl 17	. . . . . . . 8 bits bits sadd sadd ..^ bitsbits sadd sadd ..^
137		f1ocnvfv2 6533	. . . . . . . . 9 bits $/\$ sadd sadd ..^ bits bits sadd sadd ..^ sadd sadd ..^
138	11, 126, 137	sylancr 695	. . . . . . . 8 bits bits sadd sadd ..^ sadd sadd ..^
139	136, 138	eqtr3d 2658	. . . . . . 7 bitsbits sadd sadd ..^ sadd sadd ..^
140	139, 122	syl6eqss 3655	. . . . . 6 bitsbits sadd sadd ..^ ..^
141	128	nn0zd 11480	. . . . . . 7 bits sadd sadd ..^
142		bitsfzo 15157	. . . . . . 7 bits sadd sadd ..^ $/\$ bits sadd sadd ..^ ..^ bitsbits sadd sadd ..^ ..^
143	141, 60, 142	syl2anc 693	. . . . . 6 bits sadd sadd ..^ ..^ bitsbits sadd sadd ..^ ..^
144	140, 143	mpbird 247	. . . . 5 bits sadd sadd ..^ ..^
145		elfzolt2 12479	. . . . 5 bits sadd sadd ..^ ..^ bits sadd sadd ..^
146	144, 145	syl 17	. . . 4 bits sadd sadd ..^
147		modid 12695	. . . 4 bits sadd sadd ..^ $/\$ $/\$ bits sadd sadd ..^ $/\$ bits sadd sadd ..^ bits sadd sadd ..^ bits sadd sadd ..^
148	129, 133, 134, 146, 147	syl22anc 1327	. . 3 bits sadd sadd ..^ bits sadd sadd ..^
149	90, 117, 148	3eqtr3d 2664	. 2 bits sadd sadd ..^ bits sadd sadd ..^
150		f1of1 6136	. . . . 5 bits bits
151	11, 12, 150	mp2b 10	. . . 4 bits
152		f1fveq 6519	. . . 4 bits $/\$ sadd sadd ..^ $/\$ sadd sadd ..^ bits sadd sadd ..^ bits sadd sadd ..^ sadd sadd ..^ sadd sadd ..^
153	151, 152	mpan 706	. . 3 sadd sadd ..^ $/\$ sadd sadd ..^ bits sadd sadd ..^ bits sadd sadd ..^ sadd sadd ..^ sadd sadd ..^
154	99, 126, 153	syl2anc 693	. 2 bits sadd sadd ..^ bits sadd sadd ..^ sadd sadd ..^ sadd sadd ..^
155	149, 154	mpbid 222	1 sadd sadd ..^ sadd sadd ..^

Colors of variables: wff setvar class

Syntax hints:

wi 4

wb 196 $/\$ wa 384

wceq 1483 caddwcad 1545

wcel 1990

cin 3573

wss 3574

c0 3915

cif 4086

cpw 4158 class class class wbr 4653

cmpt 4729

ccnv 5113

cres 5116

wf 5884

wf1 5885

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

cn 11020

c2 11070

cn0 11292

cz 11377

crp 11832 ..^cfzo 12465

cmo 12668

cseq 12801

cexp 12860 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: sadass 15193

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