ccatass - Metamath Proof Explorer

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Theorem ccatass 13371

Description: Associative law for concatenation of words. (Contributed by Stefan O'Rear, 15-Aug-2015.)

**Assertion**
Ref	Expression
ccatass	Word $/\$ Word $/\$ Word ++ ++ ++ ++

Proof of Theorem ccatass Dummy variable

is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ccatcl 13359	. . . . 5 Word $/\$ Word ++ Word
2		ccatcl 13359	. . . . 5 ++ Word $/\$ Word ++ ++ Word
3	1, 2	stoic3 1701	. . . 4 Word $/\$ Word $/\$ Word ++ ++ Word
4		wrdf 13310	. . . 4 ++ ++ Word ++ ++ ..^ ++ ++
5		ffn 6045	. . . 4 ++ ++ ..^ ++ ++ ++ ++ ..^ ++ ++
6	3, 4, 5	3syl 18	. . 3 Word $/\$ Word $/\$ Word ++ ++ ..^ ++ ++
7		ccatlen 13360	. . . . . . 7 ++ Word $/\$ Word ++ ++ ++
8	1, 7	stoic3 1701	. . . . . 6 Word $/\$ Word $/\$ Word ++ ++ ++
9		ccatlen 13360	. . . . . . . 8 Word $/\$ Word ++
10	9	3adant3 1081	. . . . . . 7 Word $/\$ Word $/\$ Word ++
11	10	oveq1d 6665	. . . . . 6 Word $/\$ Word $/\$ Word ++
12	8, 11	eqtrd 2656	. . . . 5 Word $/\$ Word $/\$ Word ++ ++
13	12	oveq2d 6666	. . . 4 Word $/\$ Word $/\$ Word ..^ ++ ++ ..^
14	13	fneq2d 5982	. . 3 Word $/\$ Word $/\$ Word ++ ++ ..^ ++ ++ ++ ++ ..^
15	6, 14	mpbid 222	. 2 Word $/\$ Word $/\$ Word ++ ++ ..^
16		simp1 1061	. . . . 5 Word $/\$ Word $/\$ Word Word
17		ccatcl 13359	. . . . . 6 Word $/\$ Word ++ Word
18	17	3adant1 1079	. . . . 5 Word $/\$ Word $/\$ Word ++ Word
19		ccatcl 13359	. . . . 5 Word $/\$ ++ Word ++ ++ Word
20	16, 18, 19	syl2anc 693	. . . 4 Word $/\$ Word $/\$ Word ++ ++ Word
21		wrdf 13310	. . . 4 ++ ++ Word ++ ++ ..^ ++ ++
22		ffn 6045	. . . 4 ++ ++ ..^ ++ ++ ++ ++ ..^ ++ ++
23	20, 21, 22	3syl 18	. . 3 Word $/\$ Word $/\$ Word ++ ++ ..^ ++ ++
24		ccatlen 13360	. . . . . . . 8 Word $/\$ Word ++
25	24	3adant1 1079	. . . . . . 7 Word $/\$ Word $/\$ Word ++
26	25	oveq2d 6666	. . . . . 6 Word $/\$ Word $/\$ Word ++
27		ccatlen 13360	. . . . . . 7 Word $/\$ ++ Word ++ ++ ++
28	16, 18, 27	syl2anc 693	. . . . . 6 Word $/\$ Word $/\$ Word ++ ++ ++
29		lencl 13324	. . . . . . . . 9 Word
30	29	3ad2ant1 1082	. . . . . . . 8 Word $/\$ Word $/\$ Word
31	30	nn0cnd 11353	. . . . . . 7 Word $/\$ Word $/\$ Word
32		lencl 13324	. . . . . . . . 9 Word
33	32	3ad2ant2 1083	. . . . . . . 8 Word $/\$ Word $/\$ Word
34	33	nn0cnd 11353	. . . . . . 7 Word $/\$ Word $/\$ Word
35		lencl 13324	. . . . . . . . 9 Word
36	35	3ad2ant3 1084	. . . . . . . 8 Word $/\$ Word $/\$ Word
37	36	nn0cnd 11353	. . . . . . 7 Word $/\$ Word $/\$ Word
38	31, 34, 37	addassd 10062	. . . . . 6 Word $/\$ Word $/\$ Word
39	26, 28, 38	3eqtr4d 2666	. . . . 5 Word $/\$ Word $/\$ Word ++ ++
40	39	oveq2d 6666	. . . 4 Word $/\$ Word $/\$ Word ..^ ++ ++ ..^
41	40	fneq2d 5982	. . 3 Word $/\$ Word $/\$ Word ++ ++ ..^ ++ ++ ++ ++ ..^
42	23, 41	mpbid 222	. 2 Word $/\$ Word $/\$ Word ++ ++ ..^
43	30	nn0zd 11480	. . . 4 Word $/\$ Word $/\$ Word
44		fzospliti 12500	. . . . 5 ..^ $/\$ ..^ $\/$ ..^
45	44	ancoms 469	. . . 4 $/\$ ..^ ..^ $\/$ ..^
46	43, 45	sylan 488	. . 3 Word $/\$ Word $/\$ Word $/\$ ..^ ..^ $\/$ ..^
47		simpl1 1064	. . . . . 6 Word $/\$ Word $/\$ Word $/\$ ..^ Word
48		simpl2 1065	. . . . . 6 Word $/\$ Word $/\$ Word $/\$ ..^ Word
49		simpr 477	. . . . . 6 Word $/\$ Word $/\$ Word $/\$ ..^ ..^
50		ccatval1 13361	. . . . . 6 Word $/\$ Word $/\$ ..^ ++
51	47, 48, 49, 50	syl3anc 1326	. . . . 5 Word $/\$ Word $/\$ Word $/\$ ..^ ++
52	1	3adant3 1081	. . . . . . 7 Word $/\$ Word $/\$ Word ++ Word
53	52	adantr 481	. . . . . 6 Word $/\$ Word $/\$ Word $/\$ ..^ ++ Word
54		simpl3 1066	. . . . . 6 Word $/\$ Word $/\$ Word $/\$ ..^ Word
55		uzid 11702	. . . . . . . . . . 11
56	43, 55	syl 17	. . . . . . . . . 10 Word $/\$ Word $/\$ Word
57		uzaddcl 11744	. . . . . . . . . 10 $/\$
58	56, 33, 57	syl2anc 693	. . . . . . . . 9 Word $/\$ Word $/\$ Word
59		fzoss2 12496	. . . . . . . . 9 ..^ ..^
60	58, 59	syl 17	. . . . . . . 8 Word $/\$ Word $/\$ Word ..^ ..^
61	10	oveq2d 6666	. . . . . . . 8 Word $/\$ Word $/\$ Word ..^ ++ ..^
62	60, 61	sseqtr4d 3642	. . . . . . 7 Word $/\$ Word $/\$ Word ..^ ..^ ++
63	62	sselda 3603	. . . . . 6 Word $/\$ Word $/\$ Word $/\$ ..^ ..^ ++
64		ccatval1 13361	. . . . . 6 ++ Word $/\$ Word $/\$ ..^ ++ ++ ++ ++
65	53, 54, 63, 64	syl3anc 1326	. . . . 5 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++ ++
66	18	adantr 481	. . . . . 6 Word $/\$ Word $/\$ Word $/\$ ..^ ++ Word
67		ccatval1 13361	. . . . . 6 Word $/\$ ++ Word $/\$ ..^ ++ ++
68	47, 66, 49, 67	syl3anc 1326	. . . . 5 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++
69	51, 65, 68	3eqtr4d 2666	. . . 4 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++ ++ ++
70	33	nn0zd 11480	. . . . . . 7 Word $/\$ Word $/\$ Word
71	43, 70	zaddcld 11486	. . . . . 6 Word $/\$ Word $/\$ Word
72		fzospliti 12500	. . . . . . 7 ..^ $/\$ ..^ $\/$ ..^
73	72	ancoms 469	. . . . . 6 $/\$ ..^ ..^ $\/$ ..^
74	71, 73	sylan 488	. . . . 5 Word $/\$ Word $/\$ Word $/\$ ..^ ..^ $\/$ ..^
75		simpl1 1064	. . . . . . . . 9 Word $/\$ Word $/\$ Word $/\$ ..^ Word
76		simpl2 1065	. . . . . . . . 9 Word $/\$ Word $/\$ Word $/\$ ..^ Word
77		simpr 477	. . . . . . . . 9 Word $/\$ Word $/\$ Word $/\$ ..^ ..^
78		ccatval2 13362	. . . . . . . . 9 Word $/\$ Word $/\$ ..^ ++
79	75, 76, 77, 78	syl3anc 1326	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ++
80		simpl3 1066	. . . . . . . . 9 Word $/\$ Word $/\$ Word $/\$ ..^ Word
81		fzosubel3 12528	. . . . . . . . . . 11 ..^ $/\$ ..^
82	81	ancoms 469	. . . . . . . . . 10 $/\$ ..^ ..^
83	70, 82	sylan 488	. . . . . . . . 9 Word $/\$ Word $/\$ Word $/\$ ..^ ..^
84		ccatval1 13361	. . . . . . . . 9 Word $/\$ Word $/\$ ..^ ++
85	76, 80, 83, 84	syl3anc 1326	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ++
86	79, 85	eqtr4d 2659	. . . . . . 7 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++
87	52	adantr 481	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ++ Word
88		fzoss1 12495	. . . . . . . . . . . 12 ..^ ..^
89		nn0uz 11722	. . . . . . . . . . . 12
90	88, 89	eleq2s 2719	. . . . . . . . . . 11 ..^ ..^
91	30, 90	syl 17	. . . . . . . . . 10 Word $/\$ Word $/\$ Word ..^ ..^
92	91, 61	sseqtr4d 3642	. . . . . . . . 9 Word $/\$ Word $/\$ Word ..^ ..^ ++
93	92	sselda 3603	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ..^ ++
94	87, 80, 93, 64	syl3anc 1326	. . . . . . 7 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++ ++
95	18	adantr 481	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ++ Word
96		uzid 11702	. . . . . . . . . . . . 13
97	71, 96	syl 17	. . . . . . . . . . . 12 Word $/\$ Word $/\$ Word
98		uzaddcl 11744	. . . . . . . . . . . 12 $/\$
99	97, 36, 98	syl2anc 693	. . . . . . . . . . 11 Word $/\$ Word $/\$ Word
100		fzoss2 12496	. . . . . . . . . . 11 ..^ ..^
101	99, 100	syl 17	. . . . . . . . . 10 Word $/\$ Word $/\$ Word ..^ ..^
102	26, 38	eqtr4d 2659	. . . . . . . . . . 11 Word $/\$ Word $/\$ Word ++
103	102	oveq2d 6666	. . . . . . . . . 10 Word $/\$ Word $/\$ Word ..^ ++ ..^
104	101, 103	sseqtr4d 3642	. . . . . . . . 9 Word $/\$ Word $/\$ Word ..^ ..^ ++
105	104	sselda 3603	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ..^ ++
106		ccatval2 13362	. . . . . . . 8 Word $/\$ ++ Word $/\$ ..^ ++ ++ ++ ++
107	75, 95, 105, 106	syl3anc 1326	. . . . . . 7 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++ ++
108	86, 94, 107	3eqtr4d 2666	. . . . . 6 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++ ++ ++
109	10	oveq2d 6666	. . . . . . . . . . 11 Word $/\$ Word $/\$ Word ++
110	109	adantr 481	. . . . . . . . . 10 Word $/\$ Word $/\$ Word $/\$ ..^ ++
111		elfzoelz 12470	. . . . . . . . . . . . 13 ..^
112	111	zcnd 11483	. . . . . . . . . . . 12 ..^
113	112	adantl 482	. . . . . . . . . . 11 Word $/\$ Word $/\$ Word $/\$ ..^
114	31	adantr 481	. . . . . . . . . . 11 Word $/\$ Word $/\$ Word $/\$ ..^
115	34	adantr 481	. . . . . . . . . . 11 Word $/\$ Word $/\$ Word $/\$ ..^
116	113, 114, 115	subsub4d 10423	. . . . . . . . . 10 Word $/\$ Word $/\$ Word $/\$ ..^
117	110, 116	eqtr4d 2659	. . . . . . . . 9 Word $/\$ Word $/\$ Word $/\$ ..^ ++
118	117	fveq2d 6195	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ++
119		simpl2 1065	. . . . . . . . 9 Word $/\$ Word $/\$ Word $/\$ ..^ Word
120		simpl3 1066	. . . . . . . . 9 Word $/\$ Word $/\$ Word $/\$ ..^ Word
121	38	oveq2d 6666	. . . . . . . . . . . 12 Word $/\$ Word $/\$ Word ..^ ..^
122	121	eleq2d 2687	. . . . . . . . . . 11 Word $/\$ Word $/\$ Word ..^ ..^
123	122	biimpa 501	. . . . . . . . . 10 Word $/\$ Word $/\$ Word $/\$ ..^ ..^
124	43	adantr 481	. . . . . . . . . 10 Word $/\$ Word $/\$ Word $/\$ ..^
125	70	adantr 481	. . . . . . . . . 10 Word $/\$ Word $/\$ Word $/\$ ..^
126	36	nn0zd 11480	. . . . . . . . . . . 12 Word $/\$ Word $/\$ Word
127	70, 126	zaddcld 11486	. . . . . . . . . . 11 Word $/\$ Word $/\$ Word
128	127	adantr 481	. . . . . . . . . 10 Word $/\$ Word $/\$ Word $/\$ ..^
129		fzosubel2 12527	. . . . . . . . . 10 ..^ $/\$ $/\$ $/\$ ..^
130	123, 124, 125, 128, 129	syl13anc 1328	. . . . . . . . 9 Word $/\$ Word $/\$ Word $/\$ ..^ ..^
131		ccatval2 13362	. . . . . . . . 9 Word $/\$ Word $/\$ ..^ ++
132	119, 120, 130, 131	syl3anc 1326	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ++
133	118, 132	eqtr4d 2659	. . . . . . 7 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++
134	52	adantr 481	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ++ Word
135	10, 11	oveq12d 6668	. . . . . . . . . 10 Word $/\$ Word $/\$ Word ++ ..^ ++ ..^
136	135	eleq2d 2687	. . . . . . . . 9 Word $/\$ Word $/\$ Word ++ ..^ ++ ..^
137	136	biimpar 502	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ..^ ++
138		ccatval2 13362	. . . . . . . 8 ++ Word $/\$ Word $/\$ ++ ..^ ++ ++ ++ ++
139	134, 120, 137, 138	syl3anc 1326	. . . . . . 7 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++ ++
140		simpl1 1064	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ Word
141	18	adantr 481	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ++ Word
142		fzoss1 12495	. . . . . . . . . . 11 ..^ ..^
143	58, 142	syl 17	. . . . . . . . . 10 Word $/\$ Word $/\$ Word ..^ ..^
144	143, 103	sseqtr4d 3642	. . . . . . . . 9 Word $/\$ Word $/\$ Word ..^ ..^ ++
145	144	sselda 3603	. . . . . . . 8 Word $/\$ Word $/\$ Word $/\$ ..^ ..^ ++
146	140, 141, 145, 106	syl3anc 1326	. . . . . . 7 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++ ++
147	133, 139, 146	3eqtr4d 2666	. . . . . 6 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++ ++ ++
148	108, 147	jaodan 826	. . . . 5 Word $/\$ Word $/\$ Word $/\$ ..^ $\/$ ..^ ++ ++ ++ ++
149	74, 148	syldan 487	. . . 4 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++ ++ ++
150	69, 149	jaodan 826	. . 3 Word $/\$ Word $/\$ Word $/\$ ..^ $\/$ ..^ ++ ++ ++ ++
151	46, 150	syldan 487	. 2 Word $/\$ Word $/\$ Word $/\$ ..^ ++ ++ ++ ++
152	15, 42, 151	eqfnfvd 6314	1 Word $/\$ Word $/\$ Word ++ ++ ++ ++

Colors of variables: wff setvar class

Syntax hints:

wi 4 $\/$ wo 383 $/\$ wa 384 $/\$ w3a 1037

wceq 1483

wcel 1990

wss 3574

wfn 5883

wf 5884

cfv 5888 (class class class)co 6650

cc 9934

cc0 9936

caddc 9939

cmin 10266

cn0 11292

cz 11377

cuz 11687 ..^cfzo 12465

chash 13117 Word cword 13291 ++ cconcat 13293

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-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

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 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-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-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-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-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-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-concat 13301

This theorem is referenced by: ccatw2s1ass 13407 cats1cat 13606 cats2cat 13607 frmdmnd 17396 efginvrel2 18140 efgredleme 18156 efgredlemc 18158 efgcpbllemb 18168 numclwlk1lem2foalem 27222 numclwlk1lem2fo 27228 signstfvc 30651

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