Theorem ltaddnq 7780

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 7779	. . . . . . 7 |- 1Q <Q ( 1Q +Q 1Q )
2		1nq 7739	. . . . . . . 8 |- 1Q e. Q.
3		addclnq 7748	. . . . . . . . 9 |- ( ( 1Q e. Q. /\ 1Q e. Q. ) -> ( 1Q +Q 1Q ) e. Q. )
4	2, 2, 3	mp2an 430	. . . . . . . 8 |- ( 1Q +Q 1Q ) e. Q.
5		ltmnqg 7774	. . . . . . . 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 1368	. . . . . . 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 7762	. . . . . 6 |- ( B e. Q. -> ( B .Q 1Q ) = B )
9		distrnqg 7760	. . . . . . . 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 1370	. . . . . . 7 |- ( B e. Q. -> ( B .Q ( 1Q +Q 1Q ) ) = ( ( B .Q 1Q ) +Q ( B .Q 1Q ) ) )
11	8, 8	oveq12d 6106	. . . . . . 7 |- ( B e. Q. -> ( ( B .Q 1Q ) +Q ( B .Q 1Q ) ) = ( B +Q B ) )
12	10, 11	eqtrd 2271	. . . . . 6 |- ( B e. Q. -> ( B .Q ( 1Q +Q 1Q ) ) = ( B +Q B ) )
13	7, 8, 12	3brtr3d 4162	. . . . 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 7748	. . . . . . 7 |- ( ( B e. Q. /\ B e. Q. ) -> ( B +Q B ) e. Q. )
17	16	anidms 401	. . . . . 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 7773	. . . . 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 1278	. . . 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 7754	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) = ( B +Q A ) )
24		addcomnqg 7754	. . . . 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 7755	. . . . 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 6274	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q ( B +Q B ) ) = ( B +Q ( A +Q B ) ) )
29	22, 23, 28	3brtr3d 4162	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( B +Q A ) <Q ( B +Q ( A +Q B ) ) )
30		addclnq 7748	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
31		ltanqg 7773	. . 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 1278	. 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 ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4131  (class class class)co 6088   Q.cnq 7653   1Qc1q 7654   +Q cplq 7655   .Q cmq 7656   <Q cltq 7658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-eprel 4436  df-id 4440  df-iord 4513  df-on 4515  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-1o 6691  df-oadd 6695  df-omul 6696  df-er 6811  df-ec 6813  df-qs 6817  df-ni 7677  df-pli 7678  df-mi 7679  df-lti 7680  df-plpq 7717  df-mpq 7718  df-enq 7720  df-nqqs 7721  df-plqqs 7722  df-mqqs 7723  df-1nqqs 7724  df-ltnqqs 7726

This theorem is used by:  ltexnqq  7781  nsmallnqq  7785  subhalfnqq  7787  ltbtwnnqq  7788  prarloclemarch2  7792  ltexprlemm  7973  ltexprlemopl  7974  addcanprleml  7987  addcanprlemu  7988  recexprlemm  7997  cauappcvgprlemm  8018  cauappcvgprlemopl  8019  cauappcvgprlem2  8033  caucvgprlemnkj  8039  caucvgprlemnbj  8040  caucvgprlemm  8041  caucvgprlemopl  8042  caucvgprprlemnjltk  8064  caucvgprprlemopl  8070  suplocexprlemmu  8091