1smat1 - Mathbox for Thierry Arnoux

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Theorem 1smat1 29870

Description: The submatrix of the identity matrix obtained by removing the ith row and the ith column is an identity matrix. Cf. 1marepvsma1 20389. (Contributed by Thierry Arnoux, 19-Aug-2020.)

**Hypotheses**
Ref	Expression
1smat1.1	Mat
1smat1.r
1smat1.n
1smat1.i

**Assertion**
Ref	Expression
1smat1	subMat1 Mat

Proof of Theorem 1smat1 Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . 5 subMat1 subMat1
2		1smat1.n	. . . . . 6
3	2	adantr 481	. . . . 5 $/\$ $/\$
4		1smat1.i	. . . . . 6
5	4	adantr 481	. . . . 5 $/\$ $/\$
6		1smat1.r	. . . . . . . 8
7		fzfi 12771	. . . . . . . 8
8		eqid 2622	. . . . . . . . 9 Mat Mat
9		eqid 2622	. . . . . . . . 9 Mat Mat
10		1smat1.1	. . . . . . . . 9 Mat
11	8, 9, 10	mat1bas 20255	. . . . . . . 8 $/\$ Mat
12	6, 7, 11	sylancl 694	. . . . . . 7 Mat
13		eqid 2622	. . . . . . . . 9
14	8, 13	matbas2 20227	. . . . . . . 8 $/\$ Mat
15	7, 6, 14	sylancr 695	. . . . . . 7 Mat
16	12, 15	eleqtrrd 2704	. . . . . 6
17	16	adantr 481	. . . . 5 $/\$ $/\$
18		fz1ssnn 12372	. . . . . 6
19		simprl 794	. . . . . 6 $/\$ $/\$
20	18, 19	sseldi 3601	. . . . 5 $/\$ $/\$
21		simprr 796	. . . . . 6 $/\$ $/\$
22	18, 21	sseldi 3601	. . . . 5 $/\$ $/\$
23		eqidd 2623	. . . . 5 $/\$ $/\$
24		eqidd 2623	. . . . 5 $/\$ $/\$
25	1, 3, 3, 5, 5, 17, 20, 22, 23, 24	smatlem 29863	. . . 4 $/\$ $/\$ subMat1
26		eqid 2622	. . . . 5
27		eqid 2622	. . . . 5
28	7	a1i 11	. . . . 5 $/\$ $/\$
29	6	adantr 481	. . . . 5 $/\$ $/\$
30		nnuz 11723	. . . . . . . . 9
31	20, 30	syl6eleq 2711	. . . . . . . 8 $/\$ $/\$
32		fznatpl1 12395	. . . . . . . . 9 $/\$
33	3, 19, 32	syl2anc 693	. . . . . . . 8 $/\$ $/\$
34		peano2fzr 12354	. . . . . . . 8 $/\$
35	31, 33, 34	syl2anc 693	. . . . . . 7 $/\$ $/\$
36	35, 33	jca 554	. . . . . 6 $/\$ $/\$ $/\$
37		eleq1 2689	. . . . . . 7
38		eleq1 2689	. . . . . . 7
39	37, 38	ifboth 4124	. . . . . 6 $/\$
40	36, 39	syl 17	. . . . 5 $/\$ $/\$
41	22, 30	syl6eleq 2711	. . . . . . . 8 $/\$ $/\$
42		fznatpl1 12395	. . . . . . . . 9 $/\$
43	3, 21, 42	syl2anc 693	. . . . . . . 8 $/\$ $/\$
44		peano2fzr 12354	. . . . . . . 8 $/\$
45	41, 43, 44	syl2anc 693	. . . . . . 7 $/\$ $/\$
46	45, 43	jca 554	. . . . . 6 $/\$ $/\$ $/\$
47		eleq1 2689	. . . . . . 7
48		eleq1 2689	. . . . . . 7
49	47, 48	ifboth 4124	. . . . . 6 $/\$
50	46, 49	syl 17	. . . . 5 $/\$ $/\$
51	8, 26, 27, 28, 29, 40, 50, 10	mat1ov 20254	. . . 4 $/\$ $/\$
52		simpr 477	. . . . . . . . . 10 $/\$ $/\$ $/\$
53	52	iftrued 4094	. . . . . . . . 9 $/\$ $/\$ $/\$
54	53	eqeq1d 2624	. . . . . . . 8 $/\$ $/\$ $/\$
55		simpr 477	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
56	55	iftrued 4094	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
57	56	eqeq2d 2632	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
58		simpr 477	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
59	58	iffalsed 4097	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
60	59	eqeq2d 2632	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
61	20	nnred 11035	. . . . . . . . . . . . . 14 $/\$ $/\$
62	61	ad2antrr 762	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
63		fz1ssnn 12372	. . . . . . . . . . . . . . . . 17
64	63, 4	sseldi 3601	. . . . . . . . . . . . . . . 16
65	64	nnred 11035	. . . . . . . . . . . . . . 15
66	65	ad3antrrr 766	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
67	22	nnred 11035	. . . . . . . . . . . . . . . 16 $/\$ $/\$
68	67	ad2antrr 762	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
69		1red 10055	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
70	68, 69	readdcld 10069	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
71	52	adantr 481	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
72	64	nnzd 11481	. . . . . . . . . . . . . . . 16
73	72	ad3antrrr 766	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
74	22	nnzd 11481	. . . . . . . . . . . . . . . 16 $/\$ $/\$
75	74	ad2antrr 762	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
76	66, 68, 58	nltled 10187	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
77		zleltp1 11428	. . . . . . . . . . . . . . . 16 $/\$
78	77	biimpa 501	. . . . . . . . . . . . . . 15 $/\$ $/\$
79	73, 75, 76, 78	syl21anc 1325	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
80	62, 66, 70, 71, 79	lttrd 10198	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
81	62, 80	ltned 10173	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
82	81	neneqd 2799	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
83	62, 66, 68, 71, 76	ltletrd 10197	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
84	62, 83	ltned 10173	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
85	84	neneqd 2799	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
86	82, 85	2falsed 366	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
87	60, 86	bitrd 268	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
88	57, 87	pm2.61dan 832	. . . . . . . 8 $/\$ $/\$ $/\$
89	54, 88	bitrd 268	. . . . . . 7 $/\$ $/\$ $/\$
90		simpr 477	. . . . . . . . . 10 $/\$ $/\$ $/\$
91	90	iffalsed 4097	. . . . . . . . 9 $/\$ $/\$ $/\$
92	91	eqeq1d 2624	. . . . . . . 8 $/\$ $/\$ $/\$
93		simpr 477	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
94	93	iftrued 4094	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
95	94	eqeq2d 2632	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
96	67	ad2antrr 762	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
97	65	ad3antrrr 766	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
98	61	ad2antrr 762	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
99		1red 10055	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
100	98, 99	readdcld 10069	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
101	72	ad3antrrr 766	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
102	20	nnzd 11481	. . . . . . . . . . . . . . . . 17 $/\$ $/\$
103	102	ad2antrr 762	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
104	90	adantr 481	. . . . . . . . . . . . . . . . 17 $/\$ $/\$ $/\$ $/\$
105	97, 98, 104	nltled 10187	. . . . . . . . . . . . . . . 16 $/\$ $/\$ $/\$ $/\$
106		zleltp1 11428	. . . . . . . . . . . . . . . . 17 $/\$
107	106	biimpa 501	. . . . . . . . . . . . . . . 16 $/\$ $/\$
108	101, 103, 105, 107	syl21anc 1325	. . . . . . . . . . . . . . 15 $/\$ $/\$ $/\$ $/\$
109	96, 97, 100, 93, 108	lttrd 10198	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
110	96, 109	ltned 10173	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
111	110	necomd 2849	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
112	111	neneqd 2799	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
113	96, 97, 98, 93, 105	ltletrd 10197	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
114	96, 113	ltned 10173	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
115	114	necomd 2849	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
116	115	neneqd 2799	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
117	112, 116	2falsed 366	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
118	95, 117	bitrd 268	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
119		simpr 477	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
120	119	iffalsed 4097	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
121	120	eqeq2d 2632	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
122	20	nncnd 11036	. . . . . . . . . . . . 13 $/\$ $/\$
123	122	ad3antrrr 766	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
124	22	nncnd 11036	. . . . . . . . . . . . 13 $/\$ $/\$
125	124	ad3antrrr 766	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
126		1cnd 10056	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
127		simpr 477	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
128	123, 125, 126, 127	addcan2ad 10242	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
129		simpr 477	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$ $/\$
130	129	oveq1d 6665	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$
131	128, 130	impbida 877	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
132	121, 131	bitrd 268	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
133	118, 132	pm2.61dan 832	. . . . . . . 8 $/\$ $/\$ $/\$
134	92, 133	bitrd 268	. . . . . . 7 $/\$ $/\$ $/\$
135	89, 134	pm2.61dan 832	. . . . . 6 $/\$ $/\$
136	135	ifbid 4108	. . . . 5 $/\$ $/\$
137		eqid 2622	. . . . . 6 Mat Mat
138		fzfid 12772	. . . . . 6 $/\$ $/\$
139		eqid 2622	. . . . . 6 Mat Mat
140	137, 26, 27, 138, 29, 19, 21, 139	mat1ov 20254	. . . . 5 $/\$ $/\$ Mat
141	136, 140	eqtr4d 2659	. . . 4 $/\$ $/\$ Mat
142	25, 51, 141	3eqtrd 2660	. . 3 $/\$ $/\$ subMat1 Mat
143	142	ralrimivva 2971	. 2 subMat1 Mat
144	1, 2, 2, 4, 4, 16	smatrcl 29862	. . . 4 subMat1
145		elmapfn 7880	. . . 4 subMat1 subMat1
146	144, 145	syl 17	. . 3 subMat1
147		fzfi 12771	. . . . . 6
148		eqid 2622	. . . . . . 7 Mat Mat
149	137, 148, 139	mat1bas 20255	. . . . . 6 $/\$ Mat Mat
150	6, 147, 149	sylancl 694	. . . . 5 Mat Mat
151	137, 13	matbas2 20227	. . . . . 6 $/\$ Mat
152	147, 6, 151	sylancr 695	. . . . 5 Mat
153	150, 152	eleqtrrd 2704	. . . 4 Mat
154		elmapfn 7880	. . . 4 Mat Mat
155	153, 154	syl 17	. . 3 Mat
156		eqfnov2 6767	. . 3 subMat1 $/\$ Mat subMat1 Mat subMat1 Mat
157	146, 155, 156	syl2anc 693	. 2 subMat1 Mat subMat1 Mat
158	143, 157	mpbird 247	1 subMat1 Mat

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wral 2912

cif 4086 class class class wbr 4653

cxp 5112

wfn 5883

cfv 5888 (class class class)co 6650

cmap 7857

cfn 7955

cc 9934

cr 9935

c1 9937

caddc 9939

clt 10074

cle 10075

cmin 10266

cn 11020

cz 11377

cuz 11687

cfz 12326

cbs 15857

c0g 16100

cur 18501

crg 18547 Mat cmat 20213 subMat1csmat 29859

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

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-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-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 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-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-sup 8348 df-oi 8415 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-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-fzo 12466 df-seq 12802 df-hash 13118 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-hom 15966 df-cco 15967 df-0g 16102 df-gsum 16103 df-prds 16108 df-pws 16110 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-ghm 17658 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-subrg 18778 df-lmod 18865 df-lss 18933 df-sra 19172 df-rgmod 19173 df-dsmm 20076 df-frlm 20091 df-mamu 20190 df-mat 20214 df-smat 29860

This theorem is referenced by: (None)

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