Theorem ltaddnq 9796

Description: The sum of two fractions is greater than one of them. (Contributed by NM, 14-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
ltaddnq	|- ( ( A e. Q. /\ B e. Q. ) -> A <Q ( A +Q B ) )

Proof of Theorem ltaddnq

Dummy variables x y s r t are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . 3 |- ( x = A -> x = A )
2		oveq1 6657	. . 3 |- ( x = A -> ( x +Q y ) = ( A +Q y ) )
3	1, 2	breq12d 4666	. 2 |- ( x = A -> ( x <Q ( x +Q y ) <-> A <Q ( A +Q y ) ) )
4		oveq2 6658	. . 3 |- ( y = B -> ( A +Q y ) = ( A +Q B ) )
5	4	breq2d 4665	. 2 |- ( y = B -> ( A <Q ( A +Q y ) <-> A <Q ( A +Q B ) ) )
6		1lt2nq 9795	. . . . . . . 8 |- 1Q <Q ( 1Q +Q 1Q )
7		ltmnq 9794	. . . . . . . 8 |- ( y e. Q. -> ( 1Q <Q ( 1Q +Q 1Q ) <-> ( y .Q 1Q ) <Q ( y .Q ( 1Q +Q 1Q ) ) ) )
8	6, 7	mpbii 223	. . . . . . 7 |- ( y e. Q. -> ( y .Q 1Q ) <Q ( y .Q ( 1Q +Q 1Q ) ) )
9		mulidnq 9785	. . . . . . 7 |- ( y e. Q. -> ( y .Q 1Q ) = y )
10		distrnq 9783	. . . . . . . 8 |- ( y .Q ( 1Q +Q 1Q ) ) = ( ( y .Q 1Q ) +Q ( y .Q 1Q ) )
11	9, 9	oveq12d 6668	. . . . . . . 8 |- ( y e. Q. -> ( ( y .Q 1Q ) +Q ( y .Q 1Q ) ) = ( y +Q y ) )
12	10, 11	syl5eq 2668	. . . . . . 7 |- ( y e. Q. -> ( y .Q ( 1Q +Q 1Q ) ) = ( y +Q y ) )
13	8, 9, 12	3brtr3d 4684	. . . . . 6 |- ( y e. Q. -> y <Q ( y +Q y ) )
14		ltanq 9793	. . . . . 6 |- ( x e. Q. -> ( y <Q ( y +Q y ) <-> ( x +Q y ) <Q ( x +Q ( y +Q y ) ) ) )
15	13, 14	syl5ib 234	. . . . 5 |- ( x e. Q. -> ( y e. Q. -> ( x +Q y ) <Q ( x +Q ( y +Q y ) ) ) )
16	15	imp 445	. . . 4 |- ( ( x e. Q. /\ y e. Q. ) -> ( x +Q y ) <Q ( x +Q ( y +Q y ) ) )
17		addcomnq 9773	. . . 4 |- ( x +Q y ) = ( y +Q x )
18		vex 3203	. . . . 5 |- x e. _V
19		vex 3203	. . . . 5 |- y e. _V
20		addcomnq 9773	. . . . 5 |- ( r +Q s ) = ( s +Q r )
21		addassnq 9780	. . . . 5 |- ( ( r +Q s ) +Q t ) = ( r +Q ( s +Q t ) )
22	18, 19, 19, 20, 21	caov12 6862	. . . 4 |- ( x +Q ( y +Q y ) ) = ( y +Q ( x +Q y ) )
23	16, 17, 22	3brtr3g 4686	. . 3 |- ( ( x e. Q. /\ y e. Q. ) -> ( y +Q x ) <Q ( y +Q ( x +Q y ) ) )
24		ltanq 9793	. . . 4 |- ( y e. Q. -> ( x <Q ( x +Q y ) <-> ( y +Q x ) <Q ( y +Q ( x +Q y ) ) ) )
25	24	adantl 482	. . 3 |- ( ( x e. Q. /\ y e. Q. ) -> ( x <Q ( x +Q y ) <-> ( y +Q x ) <Q ( y +Q ( x +Q y ) ) ) )
26	23, 25	mpbird 247	. 2 |- ( ( x e. Q. /\ y e. Q. ) -> x <Q ( x +Q y ) )
27	3, 5, 26	vtocl2ga 3274	1 |- ( ( A e. Q. /\ B e. Q. ) -> A <Q ( A +Q B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   Q.cnq 9674   1Qc1q 9675   +Q cplq 9677   .Q cmq 9678   <Q cltq 9680

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-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-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-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-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-omul 7565  df-er 7742  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-plpq 9730  df-mpq 9731  df-ltpq 9732  df-enq 9733  df-nq 9734  df-erq 9735  df-plq 9736  df-mq 9737  df-1nq 9738  df-ltnq 9740

This theorem is referenced by:  ltexnq  9797  nsmallnq  9799  ltbtwnnq  9800  prlem934  9855  ltaddpr  9856  ltexprlem2  9859  ltexprlem4  9861