Theorem hgt750lemg 30732

Description: Lemma for the statement 7.50 of [Helfgott] p. 69. Applying a permutation T to the three factors of a product does not change the result. (Contributed by Thierry Arnoux, 1-Jan-2022.)

Hypotheses

Ref	Expression
hgt750lemg.f	|- F = ( c e. R |-> ( c o. T ) )
hgt750lemg.t	|- ( ph -> T : ( 0..^ 3 ) -1-1-onto-> ( 0..^ 3 ) )
hgt750lemg.n	|- ( ph -> N : ( 0..^ 3 ) --> NN )
hgt750lemg.l	|- ( ph -> L : NN --> RR )
hgt750lemg.1	|- ( ph -> N e. R )

Assertion

Ref	Expression
hgt750lemg	|- ( ph -> ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( ( L ` ( ( F ` N ) ` 1 ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) ) = ( ( L ` ( N ` 0 ) ) x. ( ( L ` ( N ` 1 ) ) x. ( L ` ( N ` 2 ) ) ) ) )

Distinct variable groups:   N, c   R, c   T, c   ph, c

Allowed substitution hints:   F( c)   L( c)

Proof of Theorem hgt750lemg

Dummy variables b a are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . . . 6 |- ( a = ( T ` b ) -> ( N ` a ) = ( N ` ( T ` b ) ) )
2	1	fveq2d 6195	. . . . 5 |- ( a = ( T ` b ) -> ( L ` ( N ` a ) ) = ( L ` ( N ` ( T ` b ) ) ) )
3		tpfi 8236	. . . . . 6 |- { 0 , 1 , 2 } e. Fin
4	3	a1i 11	. . . . 5 |- ( ph -> { 0 , 1 , 2 } e. Fin )
5		hgt750lemg.t	. . . . . 6 |- ( ph -> T : ( 0..^ 3 ) -1-1-onto-> ( 0..^ 3 ) )
6		fzo0to3tp 12554	. . . . . . 7 |- ( 0..^ 3 ) = { 0 , 1 , 2 }
7		f1oeq23 6130	. . . . . . 7 |- ( ( ( 0..^ 3 ) = { 0 , 1 , 2 } /\ ( 0..^ 3 ) = { 0 , 1 , 2 } ) -> ( T : ( 0..^ 3 ) -1-1-onto-> ( 0..^ 3 ) <-> T : { 0 , 1 , 2 } -1-1-onto-> { 0 , 1 , 2 } ) )
8	6, 6, 7	mp2an 708	. . . . . 6 |- ( T : ( 0..^ 3 ) -1-1-onto-> ( 0..^ 3 ) <-> T : { 0 , 1 , 2 } -1-1-onto-> { 0 , 1 , 2 } )
9	5, 8	sylib 208	. . . . 5 |- ( ph -> T : { 0 , 1 , 2 } -1-1-onto-> { 0 , 1 , 2 } )
10		eqidd 2623	. . . . 5 |- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( T ` b ) = ( T ` b ) )
11		hgt750lemg.l	. . . . . . . 8 |- ( ph -> L : NN --> RR )
12	11	adantr 481	. . . . . . 7 |- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> L : NN --> RR )
13		hgt750lemg.n	. . . . . . . . 9 |- ( ph -> N : ( 0..^ 3 ) --> NN )
14	13	adantr 481	. . . . . . . 8 |- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> N : ( 0..^ 3 ) --> NN )
15		simpr 477	. . . . . . . . 9 |- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> a e. { 0 , 1 , 2 } )
16	15, 6	syl6eleqr 2712	. . . . . . . 8 |- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> a e. ( 0..^ 3 ) )
17	14, 16	ffvelrnd 6360	. . . . . . 7 |- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> ( N ` a ) e. NN )
18	12, 17	ffvelrnd 6360	. . . . . 6 |- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> ( L ` ( N ` a ) ) e. RR )
19	18	recnd 10068	. . . . 5 |- ( ( ph /\ a e. { 0 , 1 , 2 } ) -> ( L ` ( N ` a ) ) e. CC )
20	2, 4, 9, 10, 19	fprodf1o 14676	. . . 4 |- ( ph -> prod_ a e. { 0 , 1 , 2 } ( L ` ( N ` a ) ) = prod_ b e. { 0 , 1 , 2 } ( L ` ( N ` ( T ` b ) ) ) )
21		hgt750lemg.f	. . . . . . . . . . 11 |- F = ( c e. R |-> ( c o. T ) )
22	21	a1i 11	. . . . . . . . . 10 |- ( ph -> F = ( c e. R |-> ( c o. T ) ) )
23		simpr 477	. . . . . . . . . . 11 |- ( ( ph /\ c = N ) -> c = N )
24	23	coeq1d 5283	. . . . . . . . . 10 |- ( ( ph /\ c = N ) -> ( c o. T ) = ( N o. T ) )
25		hgt750lemg.1	. . . . . . . . . 10 |- ( ph -> N e. R )
26		f1of 6137	. . . . . . . . . . . . 13 |- ( T : ( 0..^ 3 ) -1-1-onto-> ( 0..^ 3 ) -> T : ( 0..^ 3 ) --> ( 0..^ 3 ) )
27	5, 26	syl 17	. . . . . . . . . . . 12 |- ( ph -> T : ( 0..^ 3 ) --> ( 0..^ 3 ) )
28		ovexd 6680	. . . . . . . . . . . 12 |- ( ph -> ( 0..^ 3 ) e. _V )
29		fex2 7121	. . . . . . . . . . . 12 |- ( ( T : ( 0..^ 3 ) --> ( 0..^ 3 ) /\ ( 0..^ 3 ) e. _V /\ ( 0..^ 3 ) e. _V ) -> T e. _V )
30	27, 28, 28, 29	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> T e. _V )
31		coexg 7117	. . . . . . . . . . 11 |- ( ( N e. R /\ T e. _V ) -> ( N o. T ) e. _V )
32	25, 30, 31	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( N o. T ) e. _V )
33	22, 24, 25, 32	fvmptd 6288	. . . . . . . . 9 |- ( ph -> ( F ` N ) = ( N o. T ) )
34	33	adantr 481	. . . . . . . 8 |- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( F ` N ) = ( N o. T ) )
35	34	fveq1d 6193	. . . . . . 7 |- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( ( F ` N ) ` b ) = ( ( N o. T ) ` b ) )
36		f1ofun 6139	. . . . . . . . . 10 |- ( T : ( 0..^ 3 ) -1-1-onto-> ( 0..^ 3 ) -> Fun T )
37	5, 36	syl 17	. . . . . . . . 9 |- ( ph -> Fun T )
38	37	adantr 481	. . . . . . . 8 |- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> Fun T )
39		f1odm 6141	. . . . . . . . . . 11 |- ( T : { 0 , 1 , 2 } -1-1-onto-> { 0 , 1 , 2 } -> dom T = { 0 , 1 , 2 } )
40	9, 39	syl 17	. . . . . . . . . 10 |- ( ph -> dom T = { 0 , 1 , 2 } )
41	40	eleq2d 2687	. . . . . . . . 9 |- ( ph -> ( b e. dom T <-> b e. { 0 , 1 , 2 } ) )
42	41	biimpar 502	. . . . . . . 8 |- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> b e. dom T )
43		fvco 6274	. . . . . . . 8 |- ( ( Fun T /\ b e. dom T ) -> ( ( N o. T ) ` b ) = ( N ` ( T ` b ) ) )
44	38, 42, 43	syl2anc 693	. . . . . . 7 |- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( ( N o. T ) ` b ) = ( N ` ( T ` b ) ) )
45	35, 44	eqtr2d 2657	. . . . . 6 |- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( N ` ( T ` b ) ) = ( ( F ` N ) ` b ) )
46	45	fveq2d 6195	. . . . 5 |- ( ( ph /\ b e. { 0 , 1 , 2 } ) -> ( L ` ( N ` ( T ` b ) ) ) = ( L ` ( ( F ` N ) ` b ) ) )
47	46	prodeq2dv 14653	. . . 4 |- ( ph -> prod_ b e. { 0 , 1 , 2 } ( L ` ( N ` ( T ` b ) ) ) = prod_ b e. { 0 , 1 , 2 } ( L ` ( ( F ` N ) ` b ) ) )
48	20, 47	eqtr2d 2657	. . 3 |- ( ph -> prod_ b e. { 0 , 1 , 2 } ( L ` ( ( F ` N ) ` b ) ) = prod_ a e. { 0 , 1 , 2 } ( L ` ( N ` a ) ) )
49		fveq2 6191	. . . . 5 |- ( b = 0 -> ( ( F ` N ) ` b ) = ( ( F ` N ) ` 0 ) )
50	49	fveq2d 6195	. . . 4 |- ( b = 0 -> ( L ` ( ( F ` N ) ` b ) ) = ( L ` ( ( F ` N ) ` 0 ) ) )
51		fveq2 6191	. . . . 5 |- ( b = 1 -> ( ( F ` N ) ` b ) = ( ( F ` N ) ` 1 ) )
52	51	fveq2d 6195	. . . 4 |- ( b = 1 -> ( L ` ( ( F ` N ) ` b ) ) = ( L ` ( ( F ` N ) ` 1 ) ) )
53		c0ex 10034	. . . . 5 |- 0 e. _V
54	53	a1i 11	. . . 4 |- ( ph -> 0 e. _V )
55		1ex 10035	. . . . 5 |- 1 e. _V
56	55	a1i 11	. . . 4 |- ( ph -> 1 e. _V )
57	33	fveq1d 6193	. . . . . . . 8 |- ( ph -> ( ( F ` N ) ` 0 ) = ( ( N o. T ) ` 0 ) )
58	53	tpid1 4303	. . . . . . . . . 10 |- 0 e. { 0 , 1 , 2 }
59	58, 40	syl5eleqr 2708	. . . . . . . . 9 |- ( ph -> 0 e. dom T )
60		fvco 6274	. . . . . . . . 9 |- ( ( Fun T /\ 0 e. dom T ) -> ( ( N o. T ) ` 0 ) = ( N ` ( T ` 0 ) ) )
61	37, 59, 60	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( N o. T ) ` 0 ) = ( N ` ( T ` 0 ) ) )
62	57, 61	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( F ` N ) ` 0 ) = ( N ` ( T ` 0 ) ) )
63	58, 6	eleqtrri 2700	. . . . . . . . . 10 |- 0 e. ( 0..^ 3 )
64	63	a1i 11	. . . . . . . . 9 |- ( ph -> 0 e. ( 0..^ 3 ) )
65	27, 64	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( T ` 0 ) e. ( 0..^ 3 ) )
66	13, 65	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( N ` ( T ` 0 ) ) e. NN )
67	62, 66	eqeltrd 2701	. . . . . 6 |- ( ph -> ( ( F ` N ) ` 0 ) e. NN )
68	11, 67	ffvelrnd 6360	. . . . 5 |- ( ph -> ( L ` ( ( F ` N ) ` 0 ) ) e. RR )
69	68	recnd 10068	. . . 4 |- ( ph -> ( L ` ( ( F ` N ) ` 0 ) ) e. CC )
70	33	fveq1d 6193	. . . . . . . 8 |- ( ph -> ( ( F ` N ) ` 1 ) = ( ( N o. T ) ` 1 ) )
71	55	tpid2 4304	. . . . . . . . . 10 |- 1 e. { 0 , 1 , 2 }
72	71, 40	syl5eleqr 2708	. . . . . . . . 9 |- ( ph -> 1 e. dom T )
73		fvco 6274	. . . . . . . . 9 |- ( ( Fun T /\ 1 e. dom T ) -> ( ( N o. T ) ` 1 ) = ( N ` ( T ` 1 ) ) )
74	37, 72, 73	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( N o. T ) ` 1 ) = ( N ` ( T ` 1 ) ) )
75	70, 74	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( F ` N ) ` 1 ) = ( N ` ( T ` 1 ) ) )
76	71, 6	eleqtrri 2700	. . . . . . . . . 10 |- 1 e. ( 0..^ 3 )
77	76	a1i 11	. . . . . . . . 9 |- ( ph -> 1 e. ( 0..^ 3 ) )
78	27, 77	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( T ` 1 ) e. ( 0..^ 3 ) )
79	13, 78	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( N ` ( T ` 1 ) ) e. NN )
80	75, 79	eqeltrd 2701	. . . . . 6 |- ( ph -> ( ( F ` N ) ` 1 ) e. NN )
81	11, 80	ffvelrnd 6360	. . . . 5 |- ( ph -> ( L ` ( ( F ` N ) ` 1 ) ) e. RR )
82	81	recnd 10068	. . . 4 |- ( ph -> ( L ` ( ( F ` N ) ` 1 ) ) e. CC )
83		0ne1 11088	. . . . 5 |- 0 =/= 1
84	83	a1i 11	. . . 4 |- ( ph -> 0 =/= 1 )
85		fveq2 6191	. . . . 5 |- ( b = 2 -> ( ( F ` N ) ` b ) = ( ( F ` N ) ` 2 ) )
86	85	fveq2d 6195	. . . 4 |- ( b = 2 -> ( L ` ( ( F ` N ) ` b ) ) = ( L ` ( ( F ` N ) ` 2 ) ) )
87		2ex 11092	. . . . 5 |- 2 e. _V
88	87	a1i 11	. . . 4 |- ( ph -> 2 e. _V )
89	33	fveq1d 6193	. . . . . . . 8 |- ( ph -> ( ( F ` N ) ` 2 ) = ( ( N o. T ) ` 2 ) )
90	87	tpid3 4307	. . . . . . . . . 10 |- 2 e. { 0 , 1 , 2 }
91	90, 40	syl5eleqr 2708	. . . . . . . . 9 |- ( ph -> 2 e. dom T )
92		fvco 6274	. . . . . . . . 9 |- ( ( Fun T /\ 2 e. dom T ) -> ( ( N o. T ) ` 2 ) = ( N ` ( T ` 2 ) ) )
93	37, 91, 92	syl2anc 693	. . . . . . . 8 |- ( ph -> ( ( N o. T ) ` 2 ) = ( N ` ( T ` 2 ) ) )
94	89, 93	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( F ` N ) ` 2 ) = ( N ` ( T ` 2 ) ) )
95	90, 6	eleqtrri 2700	. . . . . . . . . 10 |- 2 e. ( 0..^ 3 )
96	95	a1i 11	. . . . . . . . 9 |- ( ph -> 2 e. ( 0..^ 3 ) )
97	27, 96	ffvelrnd 6360	. . . . . . . 8 |- ( ph -> ( T ` 2 ) e. ( 0..^ 3 ) )
98	13, 97	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( N ` ( T ` 2 ) ) e. NN )
99	94, 98	eqeltrd 2701	. . . . . 6 |- ( ph -> ( ( F ` N ) ` 2 ) e. NN )
100	11, 99	ffvelrnd 6360	. . . . 5 |- ( ph -> ( L ` ( ( F ` N ) ` 2 ) ) e. RR )
101	100	recnd 10068	. . . 4 |- ( ph -> ( L ` ( ( F ` N ) ` 2 ) ) e. CC )
102		0ne2 11239	. . . . 5 |- 0 =/= 2
103	102	a1i 11	. . . 4 |- ( ph -> 0 =/= 2 )
104		1ne2 11240	. . . . 5 |- 1 =/= 2
105	104	a1i 11	. . . 4 |- ( ph -> 1 =/= 2 )
106	50, 52, 54, 56, 69, 82, 84, 86, 88, 101, 103, 105	prodtp 29573	. . 3 |- ( ph -> prod_ b e. { 0 , 1 , 2 } ( L ` ( ( F ` N ) ` b ) ) = ( ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( L ` ( ( F ` N ) ` 1 ) ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) )
107		fveq2 6191	. . . . 5 |- ( a = 0 -> ( N ` a ) = ( N ` 0 ) )
108	107	fveq2d 6195	. . . 4 |- ( a = 0 -> ( L ` ( N ` a ) ) = ( L ` ( N ` 0 ) ) )
109		fveq2 6191	. . . . 5 |- ( a = 1 -> ( N ` a ) = ( N ` 1 ) )
110	109	fveq2d 6195	. . . 4 |- ( a = 1 -> ( L ` ( N ` a ) ) = ( L ` ( N ` 1 ) ) )
111	13, 64	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( N ` 0 ) e. NN )
112	11, 111	ffvelrnd 6360	. . . . 5 |- ( ph -> ( L ` ( N ` 0 ) ) e. RR )
113	112	recnd 10068	. . . 4 |- ( ph -> ( L ` ( N ` 0 ) ) e. CC )
114	13, 77	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( N ` 1 ) e. NN )
115	11, 114	ffvelrnd 6360	. . . . 5 |- ( ph -> ( L ` ( N ` 1 ) ) e. RR )
116	115	recnd 10068	. . . 4 |- ( ph -> ( L ` ( N ` 1 ) ) e. CC )
117		fveq2 6191	. . . . 5 |- ( a = 2 -> ( N ` a ) = ( N ` 2 ) )
118	117	fveq2d 6195	. . . 4 |- ( a = 2 -> ( L ` ( N ` a ) ) = ( L ` ( N ` 2 ) ) )
119	13, 96	ffvelrnd 6360	. . . . . 6 |- ( ph -> ( N ` 2 ) e. NN )
120	11, 119	ffvelrnd 6360	. . . . 5 |- ( ph -> ( L ` ( N ` 2 ) ) e. RR )
121	120	recnd 10068	. . . 4 |- ( ph -> ( L ` ( N ` 2 ) ) e. CC )
122	108, 110, 54, 56, 113, 116, 84, 118, 88, 121, 103, 105	prodtp 29573	. . 3 |- ( ph -> prod_ a e. { 0 , 1 , 2 } ( L ` ( N ` a ) ) = ( ( ( L ` ( N ` 0 ) ) x. ( L ` ( N ` 1 ) ) ) x. ( L ` ( N ` 2 ) ) ) )
123	48, 106, 122	3eqtr3d 2664	. 2 |- ( ph -> ( ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( L ` ( ( F ` N ) ` 1 ) ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) = ( ( ( L ` ( N ` 0 ) ) x. ( L ` ( N ` 1 ) ) ) x. ( L ` ( N ` 2 ) ) ) )
124	69, 82, 101	mulassd 10063	. 2 |- ( ph -> ( ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( L ` ( ( F ` N ) ` 1 ) ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) = ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( ( L ` ( ( F ` N ) ` 1 ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) ) )
125	113, 116, 121	mulassd 10063	. 2 |- ( ph -> ( ( ( L ` ( N ` 0 ) ) x. ( L ` ( N ` 1 ) ) ) x. ( L ` ( N ` 2 ) ) ) = ( ( L ` ( N ` 0 ) ) x. ( ( L ` ( N ` 1 ) ) x. ( L ` ( N ` 2 ) ) ) ) )
126	123, 124, 125	3eqtr3d 2664	1 |- ( ph -> ( ( L ` ( ( F ` N ) ` 0 ) ) x. ( ( L ` ( ( F ` N ) ` 1 ) ) x. ( L ` ( ( F ` N ) ` 2 ) ) ) ) = ( ( L ` ( N ` 0 ) ) x. ( ( L ` ( N ` 1 ) ) x. ( L ` ( N ` 2 ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   {ctp 4181   |-> cmpt 4729   dom cdm 5114   o. ccom 5118   Fun wfun 5882   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   RRcr 9935   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   2c2 11070   3c3 11071  ..^cfzo 12465   prod_cprod 14635

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-tru 1486  df-fal 1489  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-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-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  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-prod 14636

This theorem is referenced by:  hgt750lema  30735