Theorem maxltsup 11841

Description: Two ways of saying the maximum of two numbers is less than a third. (Contributed by Jim Kingdon, 10-Feb-2022.)

Assertion

Ref	Expression
maxltsup	|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { A , B } , RR , < ) < C <-> ( A < C /\ B < C ) ) )

Proof of Theorem maxltsup

Step	Hyp	Ref	Expression
1		simpl1 1027	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ sup ( { A , B } , RR , < ) < C ) -> A e. RR )
2		simpl2 1028	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ sup ( { A , B } , RR , < ) < C ) -> B e. RR )
3		maxcl 11833	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR , < ) e. RR )
4	1, 2, 3	syl2anc 411	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ sup ( { A , B } , RR , < ) < C ) -> sup ( { A , B } , RR , < ) e. RR )
5		simpl3 1029	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ sup ( { A , B } , RR , < ) < C ) -> C e. RR )
6		maxle1 11834	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> A <_ sup ( { A , B } , RR , < ) )
7	6	3adant3 1044	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> A <_ sup ( { A , B } , RR , < ) )
8	7	adantr 276	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ sup ( { A , B } , RR , < ) < C ) -> A <_ sup ( { A , B } , RR , < ) )
9		simpr 110	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ sup ( { A , B } , RR , < ) < C ) -> sup ( { A , B } , RR , < ) < C )
10	1, 4, 5, 8, 9	lelttrd 8346	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ sup ( { A , B } , RR , < ) < C ) -> A < C )
11		maxle2 11835	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> B <_ sup ( { A , B } , RR , < ) )
12	1, 2, 11	syl2anc 411	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ sup ( { A , B } , RR , < ) < C ) -> B <_ sup ( { A , B } , RR , < ) )
13	2, 4, 5, 12, 9	lelttrd 8346	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ sup ( { A , B } , RR , < ) < C ) -> B < C )
14	10, 13	jca 306	. 2 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ sup ( { A , B } , RR , < ) < C ) -> ( A < C /\ B < C ) )
15		maxabs 11832	. . . . 5 |- ( ( A e. RR /\ B e. RR ) -> sup ( { A , B } , RR , < ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
16	15	3adant3 1044	. . . 4 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> sup ( { A , B } , RR , < ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
17	16	adantr 276	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> sup ( { A , B } , RR , < ) = ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) )
18		2re 9255	. . . . . . . . . . . 12 |- 2 e. RR
19	18	a1i 9	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> 2 e. RR )
20		simpl3 1029	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> C e. RR )
21	19, 20	remulcld 8252	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( 2 x. C ) e. RR )
22	21	recnd 8250	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( 2 x. C ) e. CC )
23		simpl1 1027	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> A e. RR )
24	23	recnd 8250	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> A e. CC )
25		simpl2 1028	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> B e. RR )
26	25	recnd 8250	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> B e. CC )
27	24, 26	addcld 8241	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( A + B ) e. CC )
28	22, 27	negsubdi2d 8548	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> -u ( ( 2 x. C ) - ( A + B ) ) = ( ( A + B ) - ( 2 x. C ) ) )
29	23, 25	readdcld 8251	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( A + B ) e. RR )
30	23, 25	resubcld 8602	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( A - B ) e. RR )
31	26	2timesd 9429	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( 2 x. B ) = ( B + B ) )
32	24, 26, 26	pnncand 8571	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( A + B ) - ( A - B ) ) = ( B + B ) )
33	31, 32	eqtr4d 2267	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( 2 x. B ) = ( ( A + B ) - ( A - B ) ) )
34		2rp 9937	. . . . . . . . . . . 12 |- 2 e. RR+
35	34	a1i 9	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> 2 e. RR+ )
36		simprr 533	. . . . . . . . . . 11 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> B < C )
37	25, 20, 35, 36	ltmul2dd 10032	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( 2 x. B ) < ( 2 x. C ) )
38	33, 37	eqbrtrrd 4117	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( A + B ) - ( A - B ) ) < ( 2 x. C ) )
39	29, 30, 21, 38	ltsub23d 8772	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( A + B ) - ( 2 x. C ) ) < ( A - B ) )
40	28, 39	eqbrtrd 4115	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> -u ( ( 2 x. C ) - ( A + B ) ) < ( A - B ) )
41	24, 26, 24	nppcan3d 8559	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( A - B ) + ( A + B ) ) = ( A + A ) )
42	24	2timesd 9429	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( 2 x. A ) = ( A + A ) )
43	41, 42	eqtr4d 2267	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( A - B ) + ( A + B ) ) = ( 2 x. A ) )
44		simprl 531	. . . . . . . . . 10 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> A < C )
45	23, 20, 35, 44	ltmul2dd 10032	. . . . . . . . 9 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( 2 x. A ) < ( 2 x. C ) )
46	43, 45	eqbrtrd 4115	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( A - B ) + ( A + B ) ) < ( 2 x. C ) )
47	30, 29, 21	ltaddsubd 8767	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( ( A - B ) + ( A + B ) ) < ( 2 x. C ) <-> ( A - B ) < ( ( 2 x. C ) - ( A + B ) ) ) )
48	46, 47	mpbid 147	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( A - B ) < ( ( 2 x. C ) - ( A + B ) ) )
49	40, 48	jca 306	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( -u ( ( 2 x. C ) - ( A + B ) ) < ( A - B ) /\ ( A - B ) < ( ( 2 x. C ) - ( A + B ) ) ) )
50	21, 29	resubcld 8602	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( 2 x. C ) - ( A + B ) ) e. RR )
51	30, 50	absltd 11797	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( abs ` ( A - B ) ) < ( ( 2 x. C ) - ( A + B ) ) <-> ( -u ( ( 2 x. C ) - ( A + B ) ) < ( A - B ) /\ ( A - B ) < ( ( 2 x. C ) - ( A + B ) ) ) ) )
52	49, 51	mpbird 167	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( abs ` ( A - B ) ) < ( ( 2 x. C ) - ( A + B ) ) )
53	30	recnd 8250	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( A - B ) e. CC )
54	53	abscld 11804	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( abs ` ( A - B ) ) e. RR )
55	29, 54, 21	ltaddsub2d 8768	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) < ( 2 x. C ) <-> ( abs ` ( A - B ) ) < ( ( 2 x. C ) - ( A + B ) ) ) )
56	52, 55	mpbird 167	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( A + B ) + ( abs ` ( A - B ) ) ) < ( 2 x. C ) )
57	29, 54	readdcld 8251	. . . . 5 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( A + B ) + ( abs ` ( A - B ) ) ) e. RR )
58	57, 20, 35	ltdivmuld 10027	. . . 4 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < C <-> ( ( A + B ) + ( abs ` ( A - B ) ) ) < ( 2 x. C ) ) )
59	56, 58	mpbird 167	. . 3 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> ( ( ( A + B ) + ( abs ` ( A - B ) ) ) / 2 ) < C )
60	17, 59	eqbrtrd 4115	. 2 |- ( ( ( A e. RR /\ B e. RR /\ C e. RR ) /\ ( A < C /\ B < C ) ) -> sup ( { A , B } , RR , < ) < C )
61	14, 60	impbida 600	1 |- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( sup ( { A , B } , RR , < ) < C <-> ( A < C /\ B < C ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   {cpr 3674   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   supcsup 7224   RRcr 8074   + caddc 8078   x. cmul 8080   < clt 8256   <_ cle 8257   - cmin 8392   -ucneg 8393   / cdiv 8894   2c2 9236   RR+crp 9932   abscabs 11620

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-mulrcl 8174  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-precex 8185  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191  ax-pre-mulgt0 8192  ax-pre-mulext 8193  ax-arch 8194  ax-caucvg 8195

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-sup 7226  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-reap 8797  df-ap 8804  df-div 8895  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-n0 9445  df-z 9524  df-uz 9800  df-rp 9933  df-seqfrec 10756  df-exp 10847  df-cj 11465  df-re 11466  df-im 11467  df-rsqrt 11621  df-abs 11622

This theorem is referenced by:  ltmininf  11858  xrmaxltsup  11881  suplociccreex  15418  hovera  15441