Theorem ltaddnq 7617

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.)

Assertion

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

Proof of Theorem ltaddnq

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

Step	Hyp	Ref	Expression
1		1lt2nq 7616	. . . . . . 7 |- 1Q <Q ( 1Q +Q 1Q )
2		1nq 7576	. . . . . . . 8 |- 1Q e. Q.
3		addclnq 7585	. . . . . . . . 9 |- ( ( 1Q e. Q. /\ 1Q e. Q. ) -> ( 1Q +Q 1Q ) e. Q. )
4	2, 2, 3	mp2an 426	. . . . . . . 8 |- ( 1Q +Q 1Q ) e. Q.
5		ltmnqg 7611	. . . . . . . 8 |- ( ( 1Q e. Q. /\ ( 1Q +Q 1Q ) e. Q. /\ B e. Q. ) -> ( 1Q <Q ( 1Q +Q 1Q ) <-> ( B .Q 1Q ) <Q ( B .Q ( 1Q +Q 1Q ) ) ) )
6	2, 4, 5	mp3an12 1361	. . . . . . 7 |- ( B e. Q. -> ( 1Q <Q ( 1Q +Q 1Q ) <-> ( B .Q 1Q ) <Q ( B .Q ( 1Q +Q 1Q ) ) ) )
7	1, 6	mpbii 148	. . . . . 6 |- ( B e. Q. -> ( B .Q 1Q ) <Q ( B .Q ( 1Q +Q 1Q ) ) )
8		mulidnq 7599	. . . . . 6 |- ( B e. Q. -> ( B .Q 1Q ) = B )
9		distrnqg 7597	. . . . . . . 8 |- ( ( B e. Q. /\ 1Q e. Q. /\ 1Q e. Q. ) -> ( B .Q ( 1Q +Q 1Q ) ) = ( ( B .Q 1Q ) +Q ( B .Q 1Q ) ) )
10	2, 2, 9	mp3an23 1363	. . . . . . 7 |- ( B e. Q. -> ( B .Q ( 1Q +Q 1Q ) ) = ( ( B .Q 1Q ) +Q ( B .Q 1Q ) ) )
11	8, 8	oveq12d 6031	. . . . . . 7 |- ( B e. Q. -> ( ( B .Q 1Q ) +Q ( B .Q 1Q ) ) = ( B +Q B ) )
12	10, 11	eqtrd 2262	. . . . . 6 |- ( B e. Q. -> ( B .Q ( 1Q +Q 1Q ) ) = ( B +Q B ) )
13	7, 8, 12	3brtr3d 4117	. . . . 5 |- ( B e. Q. -> B <Q ( B +Q B ) )
14	13	adantl 277	. . . 4 |- ( ( A e. Q. /\ B e. Q. ) -> B <Q ( B +Q B ) )
15		simpr 110	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> B e. Q. )
16		addclnq 7585	. . . . . . 7 |- ( ( B e. Q. /\ B e. Q. ) -> ( B +Q B ) e. Q. )
17	16	anidms 397	. . . . . 6 |- ( B e. Q. -> ( B +Q B ) e. Q. )
18	17	adantl 277	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> ( B +Q B ) e. Q. )
19		simpl 109	. . . . 5 |- ( ( A e. Q. /\ B e. Q. ) -> A e. Q. )
20		ltanqg 7610	. . . . 5 |- ( ( B e. Q. /\ ( B +Q B ) e. Q. /\ A e. Q. ) -> ( B <Q ( B +Q B ) <-> ( A +Q B ) <Q ( A +Q ( B +Q B ) ) ) )
21	15, 18, 19, 20	syl3anc 1271	. . . 4 |- ( ( A e. Q. /\ B e. Q. ) -> ( B <Q ( B +Q B ) <-> ( A +Q B ) <Q ( A +Q ( B +Q B ) ) ) )
22	14, 21	mpbid 147	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) <Q ( A +Q ( B +Q B ) ) )
23		addcomnqg 7591	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) = ( B +Q A ) )
24		addcomnqg 7591	. . . . 5 |- ( ( r e. Q. /\ s e. Q. ) -> ( r +Q s ) = ( s +Q r ) )
25	24	adantl 277	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( r e. Q. /\ s e. Q. ) ) -> ( r +Q s ) = ( s +Q r ) )
26		addassnqg 7592	. . . . 5 |- ( ( r e. Q. /\ s e. Q. /\ t e. Q. ) -> ( ( r +Q s ) +Q t ) = ( r +Q ( s +Q t ) ) )
27	26	adantl 277	. . . 4 |- ( ( ( A e. Q. /\ B e. Q. ) /\ ( r e. Q. /\ s e. Q. /\ t e. Q. ) ) -> ( ( r +Q s ) +Q t ) = ( r +Q ( s +Q t ) ) )
28	19, 15, 15, 25, 27	caov12d 6199	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q ( B +Q B ) ) = ( B +Q ( A +Q B ) ) )
29	22, 23, 28	3brtr3d 4117	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( B +Q A ) <Q ( B +Q ( A +Q B ) ) )
30		addclnq 7585	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
31		ltanqg 7610	. . 3 |- ( ( A e. Q. /\ ( A +Q B ) e. Q. /\ B e. Q. ) -> ( A <Q ( A +Q B ) <-> ( B +Q A ) <Q ( B +Q ( A +Q B ) ) ) )
32	19, 30, 15, 31	syl3anc 1271	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( A <Q ( A +Q B ) <-> ( B +Q A ) <Q ( B +Q ( A +Q B ) ) ) )
33	29, 32	mpbird 167	1 |- ( ( A e. Q. /\ B e. Q. ) -> A <Q ( A +Q B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   class class class wbr 4086  (class class class)co 6013   Q.cnq 7490   1Qc1q 7491   +Q cplq 7492   .Q cmq 7493   <Q cltq 7495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-eprel 4384  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-1o 6577  df-oadd 6581  df-omul 6582  df-er 6697  df-ec 6699  df-qs 6703  df-ni 7514  df-pli 7515  df-mi 7516  df-lti 7517  df-plpq 7554  df-mpq 7555  df-enq 7557  df-nqqs 7558  df-plqqs 7559  df-mqqs 7560  df-1nqqs 7561  df-ltnqqs 7563

This theorem is referenced by:  ltexnqq  7618  nsmallnqq  7622  subhalfnqq  7624  ltbtwnnqq  7625  prarloclemarch2  7629  ltexprlemm  7810  ltexprlemopl  7811  addcanprleml  7824  addcanprlemu  7825  recexprlemm  7834  cauappcvgprlemm  7855  cauappcvgprlemopl  7856  cauappcvgprlem2  7870  caucvgprlemnkj  7876  caucvgprlemnbj  7877  caucvgprlemm  7878  caucvgprlemopl  7879  caucvgprprlemnjltk  7901  caucvgprprlemopl  7907  suplocexprlemmu  7928