relexpsucnnr - Metamath Proof Explorer

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Theorem relexpsucnnr 13765

Description: A reduction for relation exponentiation to the right. (Contributed by RP, 22-May-2020.)

**Assertion**
Ref	Expression
relexpsucnnr	$/\$

Proof of Theorem relexpsucnnr Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2623	. . . 4 $/\$
2		simprr 796	. . . . 5 $/\$ $/\$ $/\$
3		dmeq 5324	. . . . . . . . . . 11
4		rneq 5351	. . . . . . . . . . 11
5	3, 4	uneq12d 3768	. . . . . . . . . 10
6	5	reseq2d 5396	. . . . . . . . 9
7		eqidd 2623	. . . . . . . . . . 11
8		coeq2 5280	. . . . . . . . . . . 12
9	8	mpt2eq3dv 6721	. . . . . . . . . . 11
10		id 22	. . . . . . . . . . . 12
11	10	mpteq2dv 4745	. . . . . . . . . . 11
12	7, 9, 11	seqeq123d 12810	. . . . . . . . . 10
13	12	fveq1d 6193	. . . . . . . . 9
14	6, 13	ifeq12d 4106	. . . . . . . 8
15	14	ad2antrl 764	. . . . . . 7 $/\$ $/\$ $/\$
16	15	a1i 11	. . . . . 6 $/\$ $/\$ $/\$
17		eqeq1 2626	. . . . . . . 8
18	17	anbi2d 740	. . . . . . 7 $/\$ $/\$
19	18	anbi2d 740	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20		eqeq1 2626	. . . . . . . 8
21		fveq2 6191	. . . . . . . 8
22	20, 21	ifbieq2d 4111	. . . . . . 7
23	22	eqeq1d 2624	. . . . . 6
24	16, 19, 23	3imtr4d 283	. . . . 5 $/\$ $/\$ $/\$
25	2, 24	mpcom 38	. . . 4 $/\$ $/\$ $/\$
26		elex 3212	. . . . 5
27	26	adantr 481	. . . 4 $/\$
28		simpr 477	. . . . . 6 $/\$
29	28	peano2nnd 11037	. . . . 5 $/\$
30	29	nnnn0d 11351	. . . 4 $/\$
31		dmexg 7097	. . . . . . . 8
32		rnexg 7098	. . . . . . . 8
33		unexg 6959	. . . . . . . 8 $/\$
34	31, 32, 33	syl2anc 693	. . . . . . 7
35		resiexg 7102	. . . . . . 7
36	34, 35	syl 17	. . . . . 6
37	36	adantr 481	. . . . 5 $/\$
38		fvexd 6203	. . . . 5 $/\$
39	37, 38	ifcld 4131	. . . 4 $/\$
40	1, 25, 27, 30, 39	ovmpt2d 6788	. . 3 $/\$
41		nnne0 11053	. . . . . 6
42	41	neneqd 2799	. . . . 5
43	29, 42	syl 17	. . . 4 $/\$
44	43	iffalsed 4097	. . 3 $/\$
45		elnnuz 11724	. . . . . . 7
46	45	biimpi 206	. . . . . 6
47	46	adantl 482	. . . . 5 $/\$
48		seqp1 12816	. . . . 5
49	47, 48	syl 17	. . . 4 $/\$
50		ovex 6678	. . . . . 6
51		simpl 473	. . . . . 6 $/\$
52		eqidd 2623	. . . . . . 7
53		eqid 2622	. . . . . . 7
54	52, 53	fvmptg 6280	. . . . . 6 $/\$
55	50, 51, 54	sylancr 695	. . . . 5 $/\$
56	55	oveq2d 6666	. . . 4 $/\$
57		nfcv 2764	. . . . . . 7
58		nfcv 2764	. . . . . . 7
59		nfcv 2764	. . . . . . 7
60		nfcv 2764	. . . . . . 7
61		simpl 473	. . . . . . . 8 $/\$
62	61	coeq1d 5283	. . . . . . 7 $/\$
63	57, 58, 59, 60, 62	cbvmpt2 6734	. . . . . 6
64		oveq 6656	. . . . . 6
65	63, 64	mp1i 13	. . . . 5 $/\$
66		eqidd 2623	. . . . . 6 $/\$
67		simprl 794	. . . . . . 7 $/\$ $/\$ $/\$
68	67	coeq1d 5283	. . . . . 6 $/\$ $/\$ $/\$
69		fvexd 6203	. . . . . 6 $/\$
70		fvex 6201	. . . . . . 7
71		coexg 7117	. . . . . . 7 $/\$
72	70, 51, 71	sylancr 695	. . . . . 6 $/\$
73	66, 68, 69, 27, 72	ovmpt2d 6788	. . . . 5 $/\$
74		simpr 477	. . . . . . . . . . 11 $/\$
75	74	eqeq1d 2624	. . . . . . . . . 10 $/\$
76	6	adantr 481	. . . . . . . . . 10 $/\$
77	12	adantr 481	. . . . . . . . . . 11 $/\$
78	77, 74	fveq12d 6197	. . . . . . . . . 10 $/\$
79	75, 76, 78	ifbieq12d 4113	. . . . . . . . 9 $/\$
80	79	adantl 482	. . . . . . . 8 $/\$ $/\$ $/\$
81	28	nnnn0d 11351	. . . . . . . 8 $/\$
82	37, 69	ifcld 4131	. . . . . . . 8 $/\$
83	1, 80, 27, 81, 82	ovmpt2d 6788	. . . . . . 7 $/\$
84		nnne0 11053	. . . . . . . . . 10
85	84	adantl 482	. . . . . . . . 9 $/\$
86	85	neneqd 2799	. . . . . . . 8 $/\$
87	86	iffalsed 4097	. . . . . . 7 $/\$
88	83, 87	eqtr2d 2657	. . . . . 6 $/\$
89	88	coeq1d 5283	. . . . 5 $/\$
90	65, 73, 89	3eqtrd 2660	. . . 4 $/\$
91	49, 56, 90	3eqtrd 2660	. . 3 $/\$
92	40, 44, 91	3eqtrd 2660	. 2 $/\$
93		df-relexp 13761	. . 3
94		oveq 6656	. . . . 5
95		oveq 6656	. . . . . 6
96	95	coeq1d 5283	. . . . 5
97	94, 96	eqeq12d 2637	. . . 4
98	97	imbi2d 330	. . 3 $/\$ $/\$
99	93, 98	ax-mp 5	. 2 $/\$ $/\$
100	92, 99	mpbir 221	1 $/\$

Colors of variables: wff setvar class

Syntax hints:

wn 3

wi 4

wb 196 $/\$ wa 384

wceq 1483

wcel 1990

wne 2794

cvv 3200

cun 3572

cif 4086

cmpt 4729

cid 5023

cdm 5114

crn 5115

cres 5116

ccom 5118

cfv 5888 (class class class)co 6650

cmpt2 6652

cc0 9936

c1 9937

caddc 9939

cn 11020

cn0 11292

cuz 11687

cseq 12801

crelexp 13760

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-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-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-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 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-seq 12802 df-relexp 13761

This theorem is referenced by: relexpsucr 13769 relexpsucnnl 13772 relexpcnv 13775 relexprelg 13778 relexpnndm 13781 relexp2 37969 relexpxpnnidm 37995 relexpss1d 37997 relexpmulnn 38001 trclrelexplem 38003 relexp0a 38008 trclfvcom 38015 cotrcltrcl 38017 trclfvdecomr 38020 cotrclrcl 38034

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