Theorem addltmulALT 29305

Description: A proof readability experiment for addltmul 11268. (Contributed by Stefan Allan, 30-Oct-2010.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
addltmulALT	|- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( A + B ) < ( A x. B ) )

Proof of Theorem addltmulALT

Step	Hyp	Ref	Expression
1		simpr 477	. . . . 5 |- ( ( A e. RR /\ 2 < A ) -> 2 < A )
2		2re 11090	. . . . . . . 8 |- 2 e. RR
3	2	a1i 11	. . . . . . 7 |- ( ( A e. RR /\ 2 < A ) -> 2 e. RR )
4		simpl 473	. . . . . . 7 |- ( ( A e. RR /\ 2 < A ) -> A e. RR )
5		1re 10039	. . . . . . . 8 |- 1 e. RR
6	5	a1i 11	. . . . . . 7 |- ( ( A e. RR /\ 2 < A ) -> 1 e. RR )
7		ltsub1 10524	. . . . . . 7 |- ( ( 2 e. RR /\ A e. RR /\ 1 e. RR ) -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
8	3, 4, 6, 7	syl3anc 1326	. . . . . 6 |- ( ( A e. RR /\ 2 < A ) -> ( 2 < A <-> ( 2 - 1 ) < ( A - 1 ) ) )
9		2cn 11091	. . . . . . . . 9 |- 2 e. CC
10		ax-1cn 9994	. . . . . . . . 9 |- 1 e. CC
11		df-2 11079	. . . . . . . . . 10 |- 2 = ( 1 + 1 )
12	11	eqcomi 2631	. . . . . . . . 9 |- ( 1 + 1 ) = 2
13	9, 10, 10, 12	subaddrii 10370	. . . . . . . 8 |- ( 2 - 1 ) = 1
14	13	breq1i 4660	. . . . . . 7 |- ( ( 2 - 1 ) < ( A - 1 ) <-> 1 < ( A - 1 ) )
15	14	a1i 11	. . . . . 6 |- ( ( A e. RR /\ 2 < A ) -> ( ( 2 - 1 ) < ( A - 1 ) <-> 1 < ( A - 1 ) ) )
16	8, 15	bitrd 268	. . . . 5 |- ( ( A e. RR /\ 2 < A ) -> ( 2 < A <-> 1 < ( A - 1 ) ) )
17	1, 16	mpbid 222	. . . 4 |- ( ( A e. RR /\ 2 < A ) -> 1 < ( A - 1 ) )
18		simpr 477	. . . . 5 |- ( ( B e. RR /\ 2 < B ) -> 2 < B )
19	2	a1i 11	. . . . . . 7 |- ( ( B e. RR /\ 2 < B ) -> 2 e. RR )
20		simpl 473	. . . . . . 7 |- ( ( B e. RR /\ 2 < B ) -> B e. RR )
21	5	a1i 11	. . . . . . 7 |- ( ( B e. RR /\ 2 < B ) -> 1 e. RR )
22		ltsub1 10524	. . . . . . 7 |- ( ( 2 e. RR /\ B e. RR /\ 1 e. RR ) -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
23	19, 20, 21, 22	syl3anc 1326	. . . . . 6 |- ( ( B e. RR /\ 2 < B ) -> ( 2 < B <-> ( 2 - 1 ) < ( B - 1 ) ) )
24	13	breq1i 4660	. . . . . . 7 |- ( ( 2 - 1 ) < ( B - 1 ) <-> 1 < ( B - 1 ) )
25	24	a1i 11	. . . . . 6 |- ( ( B e. RR /\ 2 < B ) -> ( ( 2 - 1 ) < ( B - 1 ) <-> 1 < ( B - 1 ) ) )
26	23, 25	bitrd 268	. . . . 5 |- ( ( B e. RR /\ 2 < B ) -> ( 2 < B <-> 1 < ( B - 1 ) ) )
27	18, 26	mpbid 222	. . . 4 |- ( ( B e. RR /\ 2 < B ) -> 1 < ( B - 1 ) )
28	17, 27	anim12i 590	. . 3 |- ( ( ( A e. RR /\ 2 < A ) /\ ( B e. RR /\ 2 < B ) ) -> ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) )
29	28	an4s 869	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) )
30		peano2rem 10348	. . . . . . . 8 |- ( A e. RR -> ( A - 1 ) e. RR )
31		peano2rem 10348	. . . . . . . 8 |- ( B e. RR -> ( B - 1 ) e. RR )
32	30, 31	anim12i 590	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( A - 1 ) e. RR /\ ( B - 1 ) e. RR ) )
33	32	anim1i 592	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) -> ( ( ( A - 1 ) e. RR /\ ( B - 1 ) e. RR ) /\ ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) )
34		mulgt1 10882	. . . . . 6 |- ( ( ( ( A - 1 ) e. RR /\ ( B - 1 ) e. RR ) /\ ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) )
35	33, 34	syl 17	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) )
36	35	ex 450	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) )
37	36	adantr 481	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) -> 1 < ( ( A - 1 ) x. ( B - 1 ) ) ) )
38		recn 10026	. . . . . . . . 9 |- ( A e. RR -> A e. CC )
39	10	a1i 11	. . . . . . . . 9 |- ( A e. RR -> 1 e. CC )
40	38, 39	jca 554	. . . . . . . 8 |- ( A e. RR -> ( A e. CC /\ 1 e. CC ) )
41		recn 10026	. . . . . . . . 9 |- ( B e. RR -> B e. CC )
42	10	a1i 11	. . . . . . . . 9 |- ( B e. RR -> 1 e. CC )
43	41, 42	jca 554	. . . . . . . 8 |- ( B e. RR -> ( B e. CC /\ 1 e. CC ) )
44	40, 43	anim12i 590	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( A e. CC /\ 1 e. CC ) /\ ( B e. CC /\ 1 e. CC ) ) )
45		mulsub 10473	. . . . . . 7 |- ( ( ( A e. CC /\ 1 e. CC ) /\ ( B e. CC /\ 1 e. CC ) ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) )
46	44, 45	syl 17	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A - 1 ) x. ( B - 1 ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) )
47	46	breq2d 4665	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( A - 1 ) x. ( B - 1 ) ) <-> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) )
48	47	biimpd 219	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( A - 1 ) x. ( B - 1 ) ) -> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) )
49	48	adantr 481	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( 1 < ( ( A - 1 ) x. ( B - 1 ) ) -> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) )
50	10	mulid2i 10043	. . . . . . . . 9 |- ( 1 x. 1 ) = 1
51		eqcom 2629	. . . . . . . . . 10 |- ( ( 1 x. 1 ) = 1 <-> 1 = ( 1 x. 1 ) )
52	51	biimpi 206	. . . . . . . . 9 |- ( ( 1 x. 1 ) = 1 -> 1 = ( 1 x. 1 ) )
53	50, 52	mp1i 13	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> 1 = ( 1 x. 1 ) )
54	53	oveq2d 6666	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. B ) + 1 ) = ( ( A x. B ) + ( 1 x. 1 ) ) )
55		mulid1 10037	. . . . . . . . . . 11 |- ( A e. CC -> ( A x. 1 ) = A )
56		eqcom 2629	. . . . . . . . . . . 12 |- ( ( A x. 1 ) = A <-> A = ( A x. 1 ) )
57	56	biimpi 206	. . . . . . . . . . 11 |- ( ( A x. 1 ) = A -> A = ( A x. 1 ) )
58	55, 57	syl 17	. . . . . . . . . 10 |- ( A e. CC -> A = ( A x. 1 ) )
59	38, 58	syl 17	. . . . . . . . 9 |- ( A e. RR -> A = ( A x. 1 ) )
60	59	adantr 481	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> A = ( A x. 1 ) )
61		mulid1 10037	. . . . . . . . . . 11 |- ( B e. CC -> ( B x. 1 ) = B )
62	41, 61	syl 17	. . . . . . . . . 10 |- ( B e. RR -> ( B x. 1 ) = B )
63		eqcom 2629	. . . . . . . . . . 11 |- ( ( B x. 1 ) = B <-> B = ( B x. 1 ) )
64	63	biimpi 206	. . . . . . . . . 10 |- ( ( B x. 1 ) = B -> B = ( B x. 1 ) )
65	62, 64	syl 17	. . . . . . . . 9 |- ( B e. RR -> B = ( B x. 1 ) )
66	65	adantl 482	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> B = ( B x. 1 ) )
67	60, 66	oveq12d 6668	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) = ( ( A x. 1 ) + ( B x. 1 ) ) )
68	54, 67	oveq12d 6668	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A x. B ) + 1 ) - ( A + B ) ) = ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) )
69	68	breq2d 4665	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( ( A x. B ) + 1 ) - ( A + B ) ) <-> 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) ) )
70		readdcl 10019	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
71	5	a1i 11	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> 1 e. RR )
72		remulcl 10021	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
73		readdcl 10019	. . . . . . . 8 |- ( ( ( A x. B ) e. RR /\ 1 e. RR ) -> ( ( A x. B ) + 1 ) e. RR )
74	72, 71, 73	syl2anc 693	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. B ) + 1 ) e. RR )
75		ltaddsub2 10503	. . . . . . 7 |- ( ( ( A + B ) e. RR /\ 1 e. RR /\ ( ( A x. B ) + 1 ) e. RR ) -> ( ( ( A + B ) + 1 ) < ( ( A x. B ) + 1 ) <-> 1 < ( ( ( A x. B ) + 1 ) - ( A + B ) ) ) )
76	70, 71, 74, 75	syl3anc 1326	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) + 1 ) < ( ( A x. B ) + 1 ) <-> 1 < ( ( ( A x. B ) + 1 ) - ( A + B ) ) ) )
77		ltadd1 10495	. . . . . . . . 9 |- ( ( ( A + B ) e. RR /\ ( A x. B ) e. RR /\ 1 e. RR ) -> ( ( A + B ) < ( A x. B ) <-> ( ( A + B ) + 1 ) < ( ( A x. B ) + 1 ) ) )
78	70, 72, 71, 77	syl3anc 1326	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + B ) < ( A x. B ) <-> ( ( A + B ) + 1 ) < ( ( A x. B ) + 1 ) ) )
79	78	bicomd 213	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) + 1 ) < ( ( A x. B ) + 1 ) <-> ( A + B ) < ( A x. B ) ) )
80	79	biimpd 219	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( ( A + B ) + 1 ) < ( ( A x. B ) + 1 ) -> ( A + B ) < ( A x. B ) ) )
81	76, 80	sylbird 250	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( ( A x. B ) + 1 ) - ( A + B ) ) -> ( A + B ) < ( A x. B ) ) )
82	69, 81	sylbird 250	. . . 4 |- ( ( A e. RR /\ B e. RR ) -> ( 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) -> ( A + B ) < ( A x. B ) ) )
83	82	adantr 481	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( 1 < ( ( ( A x. B ) + ( 1 x. 1 ) ) - ( ( A x. 1 ) + ( B x. 1 ) ) ) -> ( A + B ) < ( A x. B ) ) )
84	37, 49, 83	3syld 60	. 2 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( ( 1 < ( A - 1 ) /\ 1 < ( B - 1 ) ) -> ( A + B ) < ( A x. B ) ) )
85	29, 84	mpd 15	1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 2 < A /\ 2 < B ) ) -> ( A + B ) < ( A x. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   - cmin 10266   2c2 11070

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-2 11079

This theorem is referenced by: (None)