Theorem isinftm 29735

Description: Express x is infinitesimal with respect to y for a structure W. (Contributed by Thierry Arnoux, 30-Jan-2018.)

Hypotheses

Ref	Expression
inftm.b	|- B = ( Base ` W )
inftm.0	|- .0. = ( 0g ` W )
inftm.x	|- .x. = (.g ` W )
inftm.l	|- .< = ( lt ` W )

Assertion

Ref	Expression
isinftm	|- ( ( W e. V /\ X e. B /\ Y e. B ) -> ( X (<<< ` W ) Y <-> ( .0. .< X /\ A. n e. NN ( n .x. X ) .< Y ) ) )

Distinct variable groups:   n, W   n, X   n, Y

Allowed substitution hints:   B( n)   .< ( n)   .x. ( n)   V( n)   .0. ( n)

Proof of Theorem isinftm

Dummy variables x w y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . . 6 |- ( x = X -> ( x e. B <-> X e. B ) )
2		eleq1 2689	. . . . . 6 |- ( y = Y -> ( y e. B <-> Y e. B ) )
3	1, 2	bi2anan9 917	. . . . 5 |- ( ( x = X /\ y = Y ) -> ( ( x e. B /\ y e. B ) <-> ( X e. B /\ Y e. B ) ) )
4		simpl 473	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> x = X )
5	4	breq2d 4665	. . . . . 6 |- ( ( x = X /\ y = Y ) -> ( .0. .< x <-> .0. .< X ) )
6	4	oveq2d 6666	. . . . . . . 8 |- ( ( x = X /\ y = Y ) -> ( n .x. x ) = ( n .x. X ) )
7		simpr 477	. . . . . . . 8 |- ( ( x = X /\ y = Y ) -> y = Y )
8	6, 7	breq12d 4666	. . . . . . 7 |- ( ( x = X /\ y = Y ) -> ( ( n .x. x ) .< y <-> ( n .x. X ) .< Y ) )
9	8	ralbidv 2986	. . . . . 6 |- ( ( x = X /\ y = Y ) -> ( A. n e. NN ( n .x. x ) .< y <-> A. n e. NN ( n .x. X ) .< Y ) )
10	5, 9	anbi12d 747	. . . . 5 |- ( ( x = X /\ y = Y ) -> ( ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) <-> ( .0. .< X /\ A. n e. NN ( n .x. X ) .< Y ) ) )
11	3, 10	anbi12d 747	. . . 4 |- ( ( x = X /\ y = Y ) -> ( ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) <-> ( ( X e. B /\ Y e. B ) /\ ( .0. .< X /\ A. n e. NN ( n .x. X ) .< Y ) ) ) )
12		eqid 2622	. . . 4 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) } = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) }
13	11, 12	brabga 4989	. . 3 |- ( ( X e. B /\ Y e. B ) -> ( X { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) } Y <-> ( ( X e. B /\ Y e. B ) /\ ( .0. .< X /\ A. n e. NN ( n .x. X ) .< Y ) ) ) )
14	13	3adant1 1079	. 2 |- ( ( W e. V /\ X e. B /\ Y e. B ) -> ( X { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) } Y <-> ( ( X e. B /\ Y e. B ) /\ ( .0. .< X /\ A. n e. NN ( n .x. X ) .< Y ) ) ) )
15		elex 3212	. . . . 5 |- ( W e. V -> W e. _V )
16	15	3ad2ant1 1082	. . . 4 |- ( ( W e. V /\ X e. B /\ Y e. B ) -> W e. _V )
17		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> ( Base ` w ) = ( Base ` W ) )
18		inftm.b	. . . . . . . . . 10 |- B = ( Base ` W )
19	17, 18	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> ( Base ` w ) = B )
20	19	eleq2d 2687	. . . . . . . 8 |- ( w = W -> ( x e. ( Base ` w ) <-> x e. B ) )
21	19	eleq2d 2687	. . . . . . . 8 |- ( w = W -> ( y e. ( Base ` w ) <-> y e. B ) )
22	20, 21	anbi12d 747	. . . . . . 7 |- ( w = W -> ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) <-> ( x e. B /\ y e. B ) ) )
23		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> ( 0g ` w ) = ( 0g ` W ) )
24		inftm.0	. . . . . . . . . 10 |- .0. = ( 0g ` W )
25	23, 24	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> ( 0g ` w ) = .0. )
26		fveq2 6191	. . . . . . . . . 10 |- ( w = W -> ( lt ` w ) = ( lt ` W ) )
27		inftm.l	. . . . . . . . . 10 |- .< = ( lt ` W )
28	26, 27	syl6eqr 2674	. . . . . . . . 9 |- ( w = W -> ( lt ` w ) = .< )
29		eqidd 2623	. . . . . . . . 9 |- ( w = W -> x = x )
30	25, 28, 29	breq123d 4667	. . . . . . . 8 |- ( w = W -> ( ( 0g ` w ) ( lt ` w ) x <-> .0. .< x ) )
31		fveq2 6191	. . . . . . . . . . . 12 |- ( w = W -> (.g ` w ) = (.g ` W ) )
32		inftm.x	. . . . . . . . . . . 12 |- .x. = (.g ` W )
33	31, 32	syl6eqr 2674	. . . . . . . . . . 11 |- ( w = W -> (.g ` w ) = .x. )
34	33	oveqd 6667	. . . . . . . . . 10 |- ( w = W -> ( n (.g ` w ) x ) = ( n .x. x ) )
35		eqidd 2623	. . . . . . . . . 10 |- ( w = W -> y = y )
36	34, 28, 35	breq123d 4667	. . . . . . . . 9 |- ( w = W -> ( ( n (.g ` w ) x ) ( lt ` w ) y <-> ( n .x. x ) .< y ) )
37	36	ralbidv 2986	. . . . . . . 8 |- ( w = W -> ( A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y <-> A. n e. NN ( n .x. x ) .< y ) )
38	30, 37	anbi12d 747	. . . . . . 7 |- ( w = W -> ( ( ( 0g ` w ) ( lt ` w ) x /\ A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y ) <-> ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) )
39	22, 38	anbi12d 747	. . . . . 6 |- ( w = W -> ( ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) /\ ( ( 0g ` w ) ( lt ` w ) x /\ A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y ) ) <-> ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) ) )
40	39	opabbidv 4716	. . . . 5 |- ( w = W -> { <. x , y >. | ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) /\ ( ( 0g ` w ) ( lt ` w ) x /\ A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y ) ) } = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) } )
41		df-inftm 29732	. . . . 5 |- <<< = ( w e. _V |-> { <. x , y >. | ( ( x e. ( Base ` w ) /\ y e. ( Base ` w ) ) /\ ( ( 0g ` w ) ( lt ` w ) x /\ A. n e. NN ( n (.g ` w ) x ) ( lt ` w ) y ) ) } )
42		fvex 6201	. . . . . . . 8 |- ( Base ` W ) e. _V
43	18, 42	eqeltri 2697	. . . . . . 7 |- B e. _V
44	43, 43	xpex 6962	. . . . . 6 |- ( B X. B ) e. _V
45		opabssxp 5193	. . . . . 6 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) } C_ ( B X. B )
46	44, 45	ssexi 4803	. . . . 5 |- { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) } e. _V
47	40, 41, 46	fvmpt 6282	. . . 4 |- ( W e. _V -> (<<< ` W ) = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) } )
48	16, 47	syl 17	. . 3 |- ( ( W e. V /\ X e. B /\ Y e. B ) -> (<<< ` W ) = { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) } )
49	48	breqd 4664	. 2 |- ( ( W e. V /\ X e. B /\ Y e. B ) -> ( X (<<< ` W ) Y <-> X { <. x , y >. | ( ( x e. B /\ y e. B ) /\ ( .0. .< x /\ A. n e. NN ( n .x. x ) .< y ) ) } Y ) )
50		3simpc 1060	. . 3 |- ( ( W e. V /\ X e. B /\ Y e. B ) -> ( X e. B /\ Y e. B ) )
51	50	biantrurd 529	. 2 |- ( ( W e. V /\ X e. B /\ Y e. B ) -> ( ( .0. .< X /\ A. n e. NN ( n .x. X ) .< Y ) <-> ( ( X e. B /\ Y e. B ) /\ ( .0. .< X /\ A. n e. NN ( n .x. X ) .< Y ) ) ) )
52	14, 49, 51	3bitr4d 300	1 |- ( ( W e. V /\ X e. B /\ Y e. B ) -> ( X (<<< ` W ) Y <-> ( .0. .< X /\ A. n e. NN ( n .x. X ) .< Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   class class class wbr 4653   {copab 4712   X. cxp 5112   ` cfv 5888  (class class class)co 6650   NNcn 11020   Basecbs 15857   0gc0g 16100   ltcplt 16941  ._gcmg 17540  <<<cinftm 29730

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-inftm 29732

This theorem is referenced by:  pnfinf  29737  isarchi2  29739