Theorem ltaddnq 7590

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 7589	. . . . . . 7 |- 1Q <Q ( 1Q +Q 1Q )
2		1nq 7549	. . . . . . . 8 |- 1Q e. Q.
3		addclnq 7558	. . . . . . . . 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 7584	. . . . . . . 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 7572	. . . . . 6 |- ( B e. Q. -> ( B .Q 1Q ) = B )
9		distrnqg 7570	. . . . . . . 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 6018	. . . . . . 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 4113	. . . . 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 7558	. . . . . . 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 7583	. . . . 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 7564	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) = ( B +Q A ) )
24		addcomnqg 7564	. . . . 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 7565	. . . . 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 6186	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q ( B +Q B ) ) = ( B +Q ( A +Q B ) ) )
29	22, 23, 28	3brtr3d 4113	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( B +Q A ) <Q ( B +Q ( A +Q B ) ) )
30		addclnq 7558	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
31		ltanqg 7583	. . 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 4082  (class class class)co 6000   Q.cnq 7463   1Qc1q 7464   +Q cplq 7465   .Q cmq 7466   <Q cltq 7468

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 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-eprel 4379  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-1o 6560  df-oadd 6564  df-omul 6565  df-er 6678  df-ec 6680  df-qs 6684  df-ni 7487  df-pli 7488  df-mi 7489  df-lti 7490  df-plpq 7527  df-mpq 7528  df-enq 7530  df-nqqs 7531  df-plqqs 7532  df-mqqs 7533  df-1nqqs 7534  df-ltnqqs 7536

This theorem is referenced by:  ltexnqq  7591  nsmallnqq  7595  subhalfnqq  7597  ltbtwnnqq  7598  prarloclemarch2  7602  ltexprlemm  7783  ltexprlemopl  7784  addcanprleml  7797  addcanprlemu  7798  recexprlemm  7807  cauappcvgprlemm  7828  cauappcvgprlemopl  7829  cauappcvgprlem2  7843  caucvgprlemnkj  7849  caucvgprlemnbj  7850  caucvgprlemm  7851  caucvgprlemopl  7852  caucvgprprlemnjltk  7874  caucvgprprlemopl  7880  suplocexprlemmu  7901