Theorem addltmulALT 12018

Description: A proof readability experiment for addltmul 7229. (Contributed by Stefan Allan, 30-Oct-2010.)

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 350	. . . . 5 |- ((A e. RR /\ 2 < A) -> 2 < A)
2		2re 7163	. . . . . . . 8 |- 2 e. RR
3	2	a1i 8	. . . . . . 7 |- ((A e. RR /\ 2 < A) -> 2 e. RR)
4		simpl 346	. . . . . . 7 |- ((A e. RR /\ 2 < A) -> A e. RR)
5		1re 6598	. . . . . . . 8 |- 1 e. RR
6	5	a1i 8	. . . . . . 7 |- ((A e. RR /\ 2 < A) -> 1 e. RR)
7		ltsub1 6851	. . . . . . 7 |- ((2 e. RR /\ A e. RR /\ 1 e. RR) -> (2 < A <-> (2 - 1) < (A - 1)))
8	3, 4, 6, 7	syl111anc 1100	. . . . . 6 |- ((A e. RR /\ 2 < A) -> (2 < A <-> (2 - 1) < (A - 1)))
9		2cn 7164	. . . . . . . . 9 |- 2 e. CC
10		ax1cn 6422	. . . . . . . . 9 |- 1 e. CC
11		df-2 7154	. . . . . . . . . 10 |- 2 = (1 + 1)
12	11	eqcomi 1888	. . . . . . . . 9 |- (1 + 1) = 2
13	9, 10, 10, 12	subaddrii 6529	. . . . . . . 8 |- (2 - 1) = 1
14	13	breq1i 3345	. . . . . . 7 |- ((2 - 1) < (A - 1) <-> 1 < (A - 1))
15	14	a1i 8	. . . . . 6 |- ((A e. RR /\ 2 < A) -> ((2 - 1) < (A - 1) <-> 1 < (A - 1)))
16	8, 15	bitrd 587	. . . . 5 |- ((A e. RR /\ 2 < A) -> (2 < A <-> 1 < (A - 1)))
17	1, 16	mpbid 212	. . . 4 |- ((A e. RR /\ 2 < A) -> 1 < (A - 1))
18		simpr 350	. . . . 5 |- ((B e. RR /\ 2 < B) -> 2 < B)
19	2	a1i 8	. . . . . . 7 |- ((B e. RR /\ 2 < B) -> 2 e. RR)
20		simpl 346	. . . . . . 7 |- ((B e. RR /\ 2 < B) -> B e. RR)
21	5	a1i 8	. . . . . . 7 |- ((B e. RR /\ 2 < B) -> 1 e. RR)
22		ltsub1 6851	. . . . . . 7 |- ((2 e. RR /\ B e. RR /\ 1 e. RR) -> (2 < B <-> (2 - 1) < (B - 1)))
23	19, 20, 21, 22	syl111anc 1100	. . . . . 6 |- ((B e. RR /\ 2 < B) -> (2 < B <-> (2 - 1) < (B - 1)))
24	13	breq1i 3345	. . . . . . 7 |- ((2 - 1) < (B - 1) <-> 1 < (B - 1))
25	24	a1i 8	. . . . . 6 |- ((B e. RR /\ 2 < B) -> ((2 - 1) < (B - 1) <-> 1 < (B - 1)))
26	23, 25	bitrd 587	. . . . 5 |- ((B e. RR /\ 2 < B) -> (2 < B <-> 1 < (B - 1)))
27	18, 26	mpbid 212	. . . 4 |- ((B e. RR /\ 2 < B) -> 1 < (B - 1))
28	17, 27	anim12i 360	. . 3 |- (((A e. RR /\ 2 < A) /\ (B e. RR /\ 2 < B)) -> (1 < (A - 1) /\ 1 < (B - 1)))
29	28	an4s 566	. 2 |- (((A e. RR /\ B e. RR) /\ (2 < A /\ 2 < B)) -> (1 < (A - 1) /\ 1 < (B - 1)))
30		peano2rem 6605	. . . . . . . 8 |- (A e. RR -> (A - 1) e. RR)
31		peano2rem 6605	. . . . . . . 8 |- (B e. RR -> (B - 1) e. RR)
32	30, 31	anim12i 360	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> ((A - 1) e. RR /\ (B - 1) e. RR))
33	32	anim1i 361	. . . . . 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 7027	. . . . . 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 12	. . . . 5 |- (((A e. RR /\ B e. RR) /\ (1 < (A - 1) /\ 1 < (B - 1))) -> 1 < ((A - 1) x. (B - 1)))
36	35	ex 402	. . . 4 |- ((A e. RR /\ B e. RR) -> ((1 < (A - 1) /\ 1 < (B - 1)) -> 1 < ((A - 1) x. (B - 1))))
37	36	adantr 425	. . 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 6466	. . . . . . . . 9 |- (A e. RR -> A e. CC)
39	10	a1i 8	. . . . . . . . 9 |- (A e. RR -> 1 e. CC)
40	38, 39	jca 310	. . . . . . . 8 |- (A e. RR -> (A e. CC /\ 1 e. CC))
41		recn 6466	. . . . . . . . 9 |- (B e. RR -> B e. CC)
42	10	a1i 8	. . . . . . . . 9 |- (B e. RR -> 1 e. CC)
43	41, 42	jca 310	. . . . . . . 8 |- (B e. RR -> (B e. CC /\ 1 e. CC))
44	40, 43	anim12i 360	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> ((A e. CC /\ 1 e. CC) /\ (B e. CC /\ 1 e. CC)))
45		mulsub 6644	. . . . . . 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 12	. . . . . 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 3350	. . . . 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 170	. . . 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 425	. . 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 6486	. . . . . . . . . 10 |- (1 x. 1) = 1
51		eqcom 1886	. . . . . . . . . . 11 |- ((1 x. 1) = 1 <-> 1 = (1 x. 1))
52	51	biimpi 168	. . . . . . . . . 10 |- ((1 x. 1) = 1 -> 1 = (1 x. 1))
53	50, 52	ax-mp 7	. . . . . . . . 9 |- 1 = (1 x. 1)
54	53	a1i 8	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> 1 = (1 x. 1))
55	54	opreq2d 4898	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> ((A x. B) + 1) = ((A x. B) + (1 x. 1)))
56		mulid1 6464	. . . . . . . . . . 11 |- (A e. CC -> (A x. 1) = A)
57		eqcom 1886	. . . . . . . . . . . 12 |- ((A x. 1) = A <-> A = (A x. 1))
58	57	biimpi 168	. . . . . . . . . . 11 |- ((A x. 1) = A -> A = (A x. 1))
59	56, 58	syl 12	. . . . . . . . . 10 |- (A e. CC -> A = (A x. 1))
60	38, 59	syl 12	. . . . . . . . 9 |- (A e. RR -> A = (A x. 1))
61	60	adantr 425	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> A = (A x. 1))
62		mulid1 6464	. . . . . . . . . . 11 |- (B e. CC -> (B x. 1) = B)
63	41, 62	syl 12	. . . . . . . . . 10 |- (B e. RR -> (B x. 1) = B)
64		eqcom 1886	. . . . . . . . . . 11 |- ((B x. 1) = B <-> B = (B x. 1))
65	64	biimpi 168	. . . . . . . . . 10 |- ((B x. 1) = B -> B = (B x. 1))
66	63, 65	syl 12	. . . . . . . . 9 |- (B e. RR -> B = (B x. 1))
67	66	adantl 424	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> B = (B x. 1))
68	61, 67	opreq12d 4900	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> (A + B) = ((A x. 1) + (B x. 1)))
69	55, 68	opreq12d 4900	. . . . . 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))))
70	69	breq2d 3350	. . . . 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)))))
71		axaddrcl 6425	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> (A + B) e. RR)
72	5	a1i 8	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> 1 e. RR)
73		axmulrcl 6427	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> (A x. B) e. RR)
74		readdcl 6455	. . . . . . . 8 |- (((A x. B) e. RR /\ 1 e. RR) -> ((A x. B) + 1) e. RR)
75	73, 72, 74	syl11anc 524	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> ((A x. B) + 1) e. RR)
76		ltaddsub2 6815	. . . . . . 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))))
77	71, 72, 75, 76	syl111anc 1100	. . . . . 6 |- ((A e. RR /\ B e. RR) -> (((A + B) + 1) < ((A x. B) + 1) <-> 1 < (((A x. B) + 1) - (A + B))))
78		ltadd1 6806	. . . . . . . . 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)))
79	71, 73, 72, 78	syl111anc 1100	. . . . . . . 8 |- ((A e. RR /\ B e. RR) -> ((A + B) < (A x. B) <-> ((A + B) + 1) < ((A x. B) + 1)))
80	79	bicomd 580	. . . . . . 7 |- ((A e. RR /\ B e. RR) -> (((A + B) + 1) < ((A x. B) + 1) <-> (A + B) < (A x. B)))
81	80	biimpd 170	. . . . . 6 |- ((A e. RR /\ B e. RR) -> (((A + B) + 1) < ((A x. B) + 1) -> (A + B) < (A x. B)))
82	77, 81	sylbird 222	. . . . 5 |- ((A e. RR /\ B e. RR) -> (1 < (((A x. B) + 1) - (A + B)) -> (A + B) < (A x. B)))
83	70, 82	sylbird 222	. . . 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)))
84	83	adantr 425	. . 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)))
85	37, 49, 84	3syld 31	. 2 |- (((A e. RR /\ B e. RR) /\ (2 < A /\ 2 < B)) -> ((1 < (A - 1) /\ 1 < (B - 1)) -> (A + B) < (A x. B)))
86	29, 85	mpd 29	1 |- (((A e. RR /\ B e. RR) /\ (2 < A /\ 2 < B)) -> (A + B) < (A x. B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300   class class class wbr 3338  (class class class)co 4884  CCcc 6384  RRcr 6385  1c1 6387   + caddc 6389   x. cmul 6391   - cmin 6445   < clt 6653  2c2 7145

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-2 7154

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