Theorem ltaddnq 7555

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 7554	. . . . . . 7 |- 1Q <Q ( 1Q +Q 1Q )
2		1nq 7514	. . . . . . . 8 |- 1Q e. Q.
3		addclnq 7523	. . . . . . . . 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 7549	. . . . . . . 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 1340	. . . . . . 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 7537	. . . . . 6 |- ( B e. Q. -> ( B .Q 1Q ) = B )
9		distrnqg 7535	. . . . . . . 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 1342	. . . . . . 7 |- ( B e. Q. -> ( B .Q ( 1Q +Q 1Q ) ) = ( ( B .Q 1Q ) +Q ( B .Q 1Q ) ) )
11	8, 8	oveq12d 5985	. . . . . . 7 |- ( B e. Q. -> ( ( B .Q 1Q ) +Q ( B .Q 1Q ) ) = ( B +Q B ) )
12	10, 11	eqtrd 2240	. . . . . 6 |- ( B e. Q. -> ( B .Q ( 1Q +Q 1Q ) ) = ( B +Q B ) )
13	7, 8, 12	3brtr3d 4090	. . . . 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 7523	. . . . . . 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 7548	. . . . 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 1250	. . . 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 7529	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) = ( B +Q A ) )
24		addcomnqg 7529	. . . . 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 7530	. . . . 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 6151	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q ( B +Q B ) ) = ( B +Q ( A +Q B ) ) )
29	22, 23, 28	3brtr3d 4090	. 2 |- ( ( A e. Q. /\ B e. Q. ) -> ( B +Q A ) <Q ( B +Q ( A +Q B ) ) )
30		addclnq 7523	. . 3 |- ( ( A e. Q. /\ B e. Q. ) -> ( A +Q B ) e. Q. )
31		ltanqg 7548	. . 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 1250	. 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 981   = wceq 1373   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   Q.cnq 7428   1Qc1q 7429   +Q cplq 7430   .Q cmq 7431   <Q cltq 7433

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-eprel 4354  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-irdg 6479  df-1o 6525  df-oadd 6529  df-omul 6530  df-er 6643  df-ec 6645  df-qs 6649  df-ni 7452  df-pli 7453  df-mi 7454  df-lti 7455  df-plpq 7492  df-mpq 7493  df-enq 7495  df-nqqs 7496  df-plqqs 7497  df-mqqs 7498  df-1nqqs 7499  df-ltnqqs 7501

This theorem is referenced by:  ltexnqq  7556  nsmallnqq  7560  subhalfnqq  7562  ltbtwnnqq  7563  prarloclemarch2  7567  ltexprlemm  7748  ltexprlemopl  7749  addcanprleml  7762  addcanprlemu  7763  recexprlemm  7772  cauappcvgprlemm  7793  cauappcvgprlemopl  7794  cauappcvgprlem2  7808  caucvgprlemnkj  7814  caucvgprlemnbj  7815  caucvgprlemm  7816  caucvgprlemopl  7817  caucvgprprlemnjltk  7839  caucvgprprlemopl  7845  suplocexprlemmu  7866