Theorem ltaddnq 7477

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 7476	. . . . . . 7 |- 1Q <Q ( 1Q +Q 1Q )
2		1nq 7436	. . . . . . . 8 |- 1Q e. Q.
3		addclnq 7445	. . . . . . . . 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 7471	. . . . . . . 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 1338	. . . . . . 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 7459	. . . . . 6 |- ( B e. Q. -> ( B .Q 1Q ) = B )
9		distrnqg 7457	. . . . . . . 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 1340	. . . . . . 7 |- ( B e. Q. -> ( B .Q ( 1Q +Q 1Q ) ) = ( ( B .Q 1Q ) +Q ( B .Q 1Q ) ) )
11	8, 8	oveq12d 5941	. . . . . . 7 |- ( B e. Q. -> ( ( B .Q 1Q ) +Q ( B .Q 1Q ) ) = ( B +Q B ) )
12	10, 11	eqtrd 2229	. . . . . 6 |- ( B e. Q. -> ( B .Q ( 1Q +Q 1Q ) ) = ( B +Q B ) )
13	7, 8, 12	3brtr3d 4065	. . . . 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 7445	. . . . . . 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 7470	. . . . 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 1249	. . . 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 7451	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) = ( B +Q A ) )
24		addcomnqg 7451	. . . . 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 7452	. . . . 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 6107	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q ( B +Q B ) ) = ( B +Q ( A +Q B ) ) )
29	22, 23, 28	3brtr3d 4065	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( B +Q A ) <Q ( B +Q ( A +Q B ) ) )
30		addclnq 7445	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
31		ltanqg 7470	. . 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 1249	. 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 980   = wceq 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5923   Q.cnq 7350   1Qc1q 7351   +Q cplq 7352   .Q cmq 7353   <Q cltq 7355

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-eprel 4325  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-ov 5926  df-oprab 5927  df-mpo 5928  df-1st 6200  df-2nd 6201  df-recs 6365  df-irdg 6430  df-1o 6476  df-oadd 6480  df-omul 6481  df-er 6594  df-ec 6596  df-qs 6600  df-ni 7374  df-pli 7375  df-mi 7376  df-lti 7377  df-plpq 7414  df-mpq 7415  df-enq 7417  df-nqqs 7418  df-plqqs 7419  df-mqqs 7420  df-1nqqs 7421  df-ltnqqs 7423

This theorem is referenced by:  ltexnqq  7478  nsmallnqq  7482  subhalfnqq  7484  ltbtwnnqq  7485  prarloclemarch2  7489  ltexprlemm  7670  ltexprlemopl  7671  addcanprleml  7684  addcanprlemu  7685  recexprlemm  7694  cauappcvgprlemm  7715  cauappcvgprlemopl  7716  cauappcvgprlem2  7730  caucvgprlemnkj  7736  caucvgprlemnbj  7737  caucvgprlemm  7738  caucvgprlemopl  7739  caucvgprprlemnjltk  7761  caucvgprprlemopl  7767  suplocexprlemmu  7788