Theorem monotuz 37506

Description: A function defined on an upper set of integers which increases at every adjacent pair is globally strictly monotonic by induction. (Contributed by Stefan O'Rear, 24-Sep-2014.)

Hypotheses

Ref	Expression
monotuz.1	|- ( ( ph /\ y e. H ) -> F < G )
monotuz.2	|- ( ( ph /\ x e. H ) -> C e. RR )
monotuz.3	|- H = ( ZZ>= ` I )
monotuz.4	|- ( x = ( y + 1 ) -> C = G )
monotuz.5	|- ( x = y -> C = F )
monotuz.6	|- ( x = A -> C = D )
monotuz.7	|- ( x = B -> C = E )

Assertion

Ref	Expression
monotuz	|- ( ( ph /\ ( A e. H /\ B e. H ) ) -> ( A < B <-> D < E ) )

Distinct variable groups:   x, A, y   x, B, y   y, C   x, D, y   x, E, y   x, F   x, G   x, H, y   ph, x, y

Allowed substitution hints:   C( x)   F( y)   G( y)   I( x, y)

Proof of Theorem monotuz

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

Step	Hyp	Ref	Expression
1		csbeq1 3536	. . 3 |- ( a = b -> [_ a / x ]_ C = [_ b / x ]_ C )
2		csbeq1 3536	. . 3 |- ( a = A -> [_ a / x ]_ C = [_ A / x ]_ C )
3		csbeq1 3536	. . 3 |- ( a = B -> [_ a / x ]_ C = [_ B / x ]_ C )
4		monotuz.3	. . . 4 |- H = ( ZZ>= ` I )
5		uzssz 11707	. . . . 5 |- ( ZZ>= ` I ) C_ ZZ
6		zssre 11384	. . . . 5 |- ZZ C_ RR
7	5, 6	sstri 3612	. . . 4 |- ( ZZ>= ` I ) C_ RR
8	4, 7	eqsstri 3635	. . 3 |- H C_ RR
9		nfv 1843	. . . . 5 |- F/ x ( ph /\ a e. H )
10		nfcsb1v 3549	. . . . . 6 |- F/_ x [_ a / x ]_ C
11	10	nfel1 2779	. . . . 5 |- F/ x [_ a / x ]_ C e. RR
12	9, 11	nfim 1825	. . . 4 |- F/ x ( ( ph /\ a e. H ) -> [_ a / x ]_ C e. RR )
13		eleq1 2689	. . . . . 6 |- ( x = a -> ( x e. H <-> a e. H ) )
14	13	anbi2d 740	. . . . 5 |- ( x = a -> ( ( ph /\ x e. H ) <-> ( ph /\ a e. H ) ) )
15		csbeq1a 3542	. . . . . 6 |- ( x = a -> C = [_ a / x ]_ C )
16	15	eleq1d 2686	. . . . 5 |- ( x = a -> ( C e. RR <-> [_ a / x ]_ C e. RR ) )
17	14, 16	imbi12d 334	. . . 4 |- ( x = a -> ( ( ( ph /\ x e. H ) -> C e. RR ) <-> ( ( ph /\ a e. H ) -> [_ a / x ]_ C e. RR ) ) )
18		monotuz.2	. . . 4 |- ( ( ph /\ x e. H ) -> C e. RR )
19	12, 17, 18	chvar 2262	. . 3 |- ( ( ph /\ a e. H ) -> [_ a / x ]_ C e. RR )
20		simpl 473	. . . . . 6 |- ( ( ( ph /\ a e. H ) /\ a < b ) -> ( ph /\ a e. H ) )
21	20	adantlrr 757	. . . . 5 |- ( ( ( ph /\ ( a e. H /\ b e. H ) ) /\ a < b ) -> ( ph /\ a e. H ) )
22	4, 5	eqsstri 3635	. . . . . . 7 |- H C_ ZZ
23		simplrl 800	. . . . . . 7 |- ( ( ( ph /\ ( a e. H /\ b e. H ) ) /\ a < b ) -> a e. H )
24	22, 23	sseldi 3601	. . . . . 6 |- ( ( ( ph /\ ( a e. H /\ b e. H ) ) /\ a < b ) -> a e. ZZ )
25		simplrr 801	. . . . . . 7 |- ( ( ( ph /\ ( a e. H /\ b e. H ) ) /\ a < b ) -> b e. H )
26	22, 25	sseldi 3601	. . . . . 6 |- ( ( ( ph /\ ( a e. H /\ b e. H ) ) /\ a < b ) -> b e. ZZ )
27		simpr 477	. . . . . 6 |- ( ( ( ph /\ ( a e. H /\ b e. H ) ) /\ a < b ) -> a < b )
28		csbeq1 3536	. . . . . . . . 9 |- ( c = ( a + 1 ) -> [_ c / x ]_ C = [_ ( a + 1 ) / x ]_ C )
29	28	breq2d 4665	. . . . . . . 8 |- ( c = ( a + 1 ) -> ( [_ a / x ]_ C < [_ c / x ]_ C <-> [_ a / x ]_ C < [_ ( a + 1 ) / x ]_ C ) )
30	29	imbi2d 330	. . . . . . 7 |- ( c = ( a + 1 ) -> ( ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ c / x ]_ C ) <-> ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ ( a + 1 ) / x ]_ C ) ) )
31		csbeq1 3536	. . . . . . . . 9 |- ( c = d -> [_ c / x ]_ C = [_ d / x ]_ C )
32	31	breq2d 4665	. . . . . . . 8 |- ( c = d -> ( [_ a / x ]_ C < [_ c / x ]_ C <-> [_ a / x ]_ C < [_ d / x ]_ C ) )
33	32	imbi2d 330	. . . . . . 7 |- ( c = d -> ( ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ c / x ]_ C ) <-> ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ d / x ]_ C ) ) )
34		csbeq1 3536	. . . . . . . . 9 |- ( c = ( d + 1 ) -> [_ c / x ]_ C = [_ ( d + 1 ) / x ]_ C )
35	34	breq2d 4665	. . . . . . . 8 |- ( c = ( d + 1 ) -> ( [_ a / x ]_ C < [_ c / x ]_ C <-> [_ a / x ]_ C < [_ ( d + 1 ) / x ]_ C ) )
36	35	imbi2d 330	. . . . . . 7 |- ( c = ( d + 1 ) -> ( ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ c / x ]_ C ) <-> ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ ( d + 1 ) / x ]_ C ) ) )
37		csbeq1 3536	. . . . . . . . 9 |- ( c = b -> [_ c / x ]_ C = [_ b / x ]_ C )
38	37	breq2d 4665	. . . . . . . 8 |- ( c = b -> ( [_ a / x ]_ C < [_ c / x ]_ C <-> [_ a / x ]_ C < [_ b / x ]_ C ) )
39	38	imbi2d 330	. . . . . . 7 |- ( c = b -> ( ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ c / x ]_ C ) <-> ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ b / x ]_ C ) ) )
40		eleq1 2689	. . . . . . . . . 10 |- ( y = a -> ( y e. H <-> a e. H ) )
41	40	anbi2d 740	. . . . . . . . 9 |- ( y = a -> ( ( ph /\ y e. H ) <-> ( ph /\ a e. H ) ) )
42		vex 3203	. . . . . . . . . . . 12 |- y e. _V
43		monotuz.5	. . . . . . . . . . . 12 |- ( x = y -> C = F )
44	42, 43	csbie 3559	. . . . . . . . . . 11 |- [_ y / x ]_ C = F
45		csbeq1 3536	. . . . . . . . . . 11 |- ( y = a -> [_ y / x ]_ C = [_ a / x ]_ C )
46	44, 45	syl5eqr 2670	. . . . . . . . . 10 |- ( y = a -> F = [_ a / x ]_ C )
47		ovex 6678	. . . . . . . . . . . 12 |- ( y + 1 ) e. _V
48		monotuz.4	. . . . . . . . . . . 12 |- ( x = ( y + 1 ) -> C = G )
49	47, 48	csbie 3559	. . . . . . . . . . 11 |- [_ ( y + 1 ) / x ]_ C = G
50		oveq1 6657	. . . . . . . . . . . 12 |- ( y = a -> ( y + 1 ) = ( a + 1 ) )
51	50	csbeq1d 3540	. . . . . . . . . . 11 |- ( y = a -> [_ ( y + 1 ) / x ]_ C = [_ ( a + 1 ) / x ]_ C )
52	49, 51	syl5eqr 2670	. . . . . . . . . 10 |- ( y = a -> G = [_ ( a + 1 ) / x ]_ C )
53	46, 52	breq12d 4666	. . . . . . . . 9 |- ( y = a -> ( F < G <-> [_ a / x ]_ C < [_ ( a + 1 ) / x ]_ C ) )
54	41, 53	imbi12d 334	. . . . . . . 8 |- ( y = a -> ( ( ( ph /\ y e. H ) -> F < G ) <-> ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ ( a + 1 ) / x ]_ C ) ) )
55		monotuz.1	. . . . . . . 8 |- ( ( ph /\ y e. H ) -> F < G )
56	54, 55	vtoclg 3266	. . . . . . 7 |- ( a e. ZZ -> ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ ( a + 1 ) / x ]_ C ) )
57	19	3ad2ant2 1083	. . . . . . . . . 10 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> [_ a / x ]_ C e. RR )
58		simp2l 1087	. . . . . . . . . . 11 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> ph )
59		zre 11381	. . . . . . . . . . . . . . . . 17 |- ( a e. ZZ -> a e. RR )
60	59	3ad2ant1 1082	. . . . . . . . . . . . . . . 16 |- ( ( a e. ZZ /\ d e. ZZ /\ a < d ) -> a e. RR )
61		zre 11381	. . . . . . . . . . . . . . . . 17 |- ( d e. ZZ -> d e. RR )
62	61	3ad2ant2 1083	. . . . . . . . . . . . . . . 16 |- ( ( a e. ZZ /\ d e. ZZ /\ a < d ) -> d e. RR )
63		simp3 1063	. . . . . . . . . . . . . . . 16 |- ( ( a e. ZZ /\ d e. ZZ /\ a < d ) -> a < d )
64	60, 62, 63	ltled 10185	. . . . . . . . . . . . . . 15 |- ( ( a e. ZZ /\ d e. ZZ /\ a < d ) -> a <_ d )
65	64	3ad2ant1 1082	. . . . . . . . . . . . . 14 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> a <_ d )
66		simp11 1091	. . . . . . . . . . . . . . 15 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> a e. ZZ )
67		simp12 1092	. . . . . . . . . . . . . . 15 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> d e. ZZ )
68		eluz 11701	. . . . . . . . . . . . . . 15 |- ( ( a e. ZZ /\ d e. ZZ ) -> ( d e. ( ZZ>= ` a ) <-> a <_ d ) )
69	66, 67, 68	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> ( d e. ( ZZ>= ` a ) <-> a <_ d ) )
70	65, 69	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> d e. ( ZZ>= ` a ) )
71		simp2r 1088	. . . . . . . . . . . . . 14 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> a e. H )
72	71, 4	syl6eleq 2711	. . . . . . . . . . . . 13 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> a e. ( ZZ>= ` I ) )
73		uztrn 11704	. . . . . . . . . . . . 13 |- ( ( d e. ( ZZ>= ` a ) /\ a e. ( ZZ>= ` I ) ) -> d e. ( ZZ>= ` I ) )
74	70, 72, 73	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> d e. ( ZZ>= ` I ) )
75	74, 4	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> d e. H )
76		nfv 1843	. . . . . . . . . . . . 13 |- F/ x ( ph /\ d e. H )
77		nfcsb1v 3549	. . . . . . . . . . . . . 14 |- F/_ x [_ d / x ]_ C
78	77	nfel1 2779	. . . . . . . . . . . . 13 |- F/ x [_ d / x ]_ C e. RR
79	76, 78	nfim 1825	. . . . . . . . . . . 12 |- F/ x ( ( ph /\ d e. H ) -> [_ d / x ]_ C e. RR )
80		eleq1 2689	. . . . . . . . . . . . . 14 |- ( x = d -> ( x e. H <-> d e. H ) )
81	80	anbi2d 740	. . . . . . . . . . . . 13 |- ( x = d -> ( ( ph /\ x e. H ) <-> ( ph /\ d e. H ) ) )
82		csbeq1a 3542	. . . . . . . . . . . . . 14 |- ( x = d -> C = [_ d / x ]_ C )
83	82	eleq1d 2686	. . . . . . . . . . . . 13 |- ( x = d -> ( C e. RR <-> [_ d / x ]_ C e. RR ) )
84	81, 83	imbi12d 334	. . . . . . . . . . . 12 |- ( x = d -> ( ( ( ph /\ x e. H ) -> C e. RR ) <-> ( ( ph /\ d e. H ) -> [_ d / x ]_ C e. RR ) ) )
85	79, 84, 18	chvar 2262	. . . . . . . . . . 11 |- ( ( ph /\ d e. H ) -> [_ d / x ]_ C e. RR )
86	58, 75, 85	syl2anc 693	. . . . . . . . . 10 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> [_ d / x ]_ C e. RR )
87		peano2uz 11741	. . . . . . . . . . . . 13 |- ( d e. ( ZZ>= ` I ) -> ( d + 1 ) e. ( ZZ>= ` I ) )
88	74, 87	syl 17	. . . . . . . . . . . 12 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> ( d + 1 ) e. ( ZZ>= ` I ) )
89	88, 4	syl6eleqr 2712	. . . . . . . . . . 11 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> ( d + 1 ) e. H )
90		nfv 1843	. . . . . . . . . . . . 13 |- F/ x ( ph /\ ( d + 1 ) e. H )
91		nfcsb1v 3549	. . . . . . . . . . . . . 14 |- F/_ x [_ ( d + 1 ) / x ]_ C
92	91	nfel1 2779	. . . . . . . . . . . . 13 |- F/ x [_ ( d + 1 ) / x ]_ C e. RR
93	90, 92	nfim 1825	. . . . . . . . . . . 12 |- F/ x ( ( ph /\ ( d + 1 ) e. H ) -> [_ ( d + 1 ) / x ]_ C e. RR )
94		ovex 6678	. . . . . . . . . . . 12 |- ( d + 1 ) e. _V
95		eleq1 2689	. . . . . . . . . . . . . 14 |- ( x = ( d + 1 ) -> ( x e. H <-> ( d + 1 ) e. H ) )
96	95	anbi2d 740	. . . . . . . . . . . . 13 |- ( x = ( d + 1 ) -> ( ( ph /\ x e. H ) <-> ( ph /\ ( d + 1 ) e. H ) ) )
97		csbeq1a 3542	. . . . . . . . . . . . . 14 |- ( x = ( d + 1 ) -> C = [_ ( d + 1 ) / x ]_ C )
98	97	eleq1d 2686	. . . . . . . . . . . . 13 |- ( x = ( d + 1 ) -> ( C e. RR <-> [_ ( d + 1 ) / x ]_ C e. RR ) )
99	96, 98	imbi12d 334	. . . . . . . . . . . 12 |- ( x = ( d + 1 ) -> ( ( ( ph /\ x e. H ) -> C e. RR ) <-> ( ( ph /\ ( d + 1 ) e. H ) -> [_ ( d + 1 ) / x ]_ C e. RR ) ) )
100	93, 94, 99, 18	vtoclf 3258	. . . . . . . . . . 11 |- ( ( ph /\ ( d + 1 ) e. H ) -> [_ ( d + 1 ) / x ]_ C e. RR )
101	58, 89, 100	syl2anc 693	. . . . . . . . . 10 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> [_ ( d + 1 ) / x ]_ C e. RR )
102		simp3 1063	. . . . . . . . . 10 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> [_ a / x ]_ C < [_ d / x ]_ C )
103		nfv 1843	. . . . . . . . . . . 12 |- F/ y ( ( ph /\ d e. H ) -> [_ d / x ]_ C < [_ ( d + 1 ) / x ]_ C )
104		eleq1 2689	. . . . . . . . . . . . . 14 |- ( y = d -> ( y e. H <-> d e. H ) )
105	104	anbi2d 740	. . . . . . . . . . . . 13 |- ( y = d -> ( ( ph /\ y e. H ) <-> ( ph /\ d e. H ) ) )
106		csbeq1 3536	. . . . . . . . . . . . . . 15 |- ( y = d -> [_ y / x ]_ C = [_ d / x ]_ C )
107	44, 106	syl5eqr 2670	. . . . . . . . . . . . . 14 |- ( y = d -> F = [_ d / x ]_ C )
108		oveq1 6657	. . . . . . . . . . . . . . . 16 |- ( y = d -> ( y + 1 ) = ( d + 1 ) )
109	108	csbeq1d 3540	. . . . . . . . . . . . . . 15 |- ( y = d -> [_ ( y + 1 ) / x ]_ C = [_ ( d + 1 ) / x ]_ C )
110	49, 109	syl5eqr 2670	. . . . . . . . . . . . . 14 |- ( y = d -> G = [_ ( d + 1 ) / x ]_ C )
111	107, 110	breq12d 4666	. . . . . . . . . . . . 13 |- ( y = d -> ( F < G <-> [_ d / x ]_ C < [_ ( d + 1 ) / x ]_ C ) )
112	105, 111	imbi12d 334	. . . . . . . . . . . 12 |- ( y = d -> ( ( ( ph /\ y e. H ) -> F < G ) <-> ( ( ph /\ d e. H ) -> [_ d / x ]_ C < [_ ( d + 1 ) / x ]_ C ) ) )
113	103, 112, 55	chvar 2262	. . . . . . . . . . 11 |- ( ( ph /\ d e. H ) -> [_ d / x ]_ C < [_ ( d + 1 ) / x ]_ C )
114	58, 75, 113	syl2anc 693	. . . . . . . . . 10 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> [_ d / x ]_ C < [_ ( d + 1 ) / x ]_ C )
115	57, 86, 101, 102, 114	lttrd 10198	. . . . . . . . 9 |- ( ( ( a e. ZZ /\ d e. ZZ /\ a < d ) /\ ( ph /\ a e. H ) /\ [_ a / x ]_ C < [_ d / x ]_ C ) -> [_ a / x ]_ C < [_ ( d + 1 ) / x ]_ C )
116	115	3exp 1264	. . . . . . . 8 |- ( ( a e. ZZ /\ d e. ZZ /\ a < d ) -> ( ( ph /\ a e. H ) -> ( [_ a / x ]_ C < [_ d / x ]_ C -> [_ a / x ]_ C < [_ ( d + 1 ) / x ]_ C ) ) )
117	116	a2d 29	. . . . . . 7 |- ( ( a e. ZZ /\ d e. ZZ /\ a < d ) -> ( ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ d / x ]_ C ) -> ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ ( d + 1 ) / x ]_ C ) ) )
118	30, 33, 36, 39, 56, 117	uzind2 11470	. . . . . 6 |- ( ( a e. ZZ /\ b e. ZZ /\ a < b ) -> ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ b / x ]_ C ) )
119	24, 26, 27, 118	syl3anc 1326	. . . . 5 |- ( ( ( ph /\ ( a e. H /\ b e. H ) ) /\ a < b ) -> ( ( ph /\ a e. H ) -> [_ a / x ]_ C < [_ b / x ]_ C ) )
120	21, 119	mpd 15	. . . 4 |- ( ( ( ph /\ ( a e. H /\ b e. H ) ) /\ a < b ) -> [_ a / x ]_ C < [_ b / x ]_ C )
121	120	ex 450	. . 3 |- ( ( ph /\ ( a e. H /\ b e. H ) ) -> ( a < b -> [_ a / x ]_ C < [_ b / x ]_ C ) )
122	1, 2, 3, 8, 19, 121	ltord1 10554	. 2 |- ( ( ph /\ ( A e. H /\ B e. H ) ) -> ( A < B <-> [_ A / x ]_ C < [_ B / x ]_ C ) )
123		nfcvd 2765	. . . . 5 |- ( A e. H -> F/_ x D )
124		monotuz.6	. . . . 5 |- ( x = A -> C = D )
125	123, 124	csbiegf 3557	. . . 4 |- ( A e. H -> [_ A / x ]_ C = D )
126		nfcvd 2765	. . . . 5 |- ( B e. H -> F/_ x E )
127		monotuz.7	. . . . 5 |- ( x = B -> C = E )
128	126, 127	csbiegf 3557	. . . 4 |- ( B e. H -> [_ B / x ]_ C = E )
129	125, 128	breqan12d 4669	. . 3 |- ( ( A e. H /\ B e. H ) -> ( [_ A / x ]_ C < [_ B / x ]_ C <-> D < E ) )
130	129	adantl 482	. 2 |- ( ( ph /\ ( A e. H /\ B e. H ) ) -> ( [_ A / x ]_ C < [_ B / x ]_ C <-> D < E ) )
131	122, 130	bitrd 268	1 |- ( ( ph /\ ( A e. H /\ B e. H ) ) -> ( A < B <-> D < E ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   [_csb 3533   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   ZZcz 11377   ZZ>=cuz 11687

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-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

This theorem is referenced by:  ltrmynn0  37515  ltrmxnn0  37516