maxltsup - Intuitionistic Logic Explorer

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Theorem maxltsup 11246

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	$/\$ $/\$ $/\$

**Proof of Theorem maxltsup**
Step	Hyp	Ref	Expression
1		simpl1 1002	. . . 4 $/\$ $/\$ $/\$
2		simpl2 1003	. . . . 5 $/\$ $/\$ $/\$
3		maxcl 11238	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
5		simpl3 1004	. . . 4 $/\$ $/\$ $/\$
6		maxle1 11239	. . . . . 6 $/\$
7	6	3adant3 1019	. . . . 5 $/\$ $/\$
8	7	adantr 276	. . . 4 $/\$ $/\$ $/\$
9		simpr 110	. . . 4 $/\$ $/\$ $/\$
10	1, 4, 5, 8, 9	lelttrd 8101	. . 3 $/\$ $/\$ $/\$
11		maxle2 11240	. . . . 5 $/\$
12	1, 2, 11	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
13	2, 4, 5, 12, 9	lelttrd 8101	. . 3 $/\$ $/\$ $/\$
14	10, 13	jca 306	. 2 $/\$ $/\$ $/\$ $/\$
15		maxabs 11237	. . . . 5 $/\$
16	15	3adant3 1019	. . . 4 $/\$ $/\$
17	16	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$
18		2re 9008	. . . . . . . . . . . 12
19	18	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
20		simpl3 1004	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	19, 20	remulcld 8007	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22	21	recnd 8005	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
23		simpl1 1002	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
24	23	recnd 8005	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
25		simpl2 1003	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
26	25	recnd 8005	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	24, 26	addcld 7996	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	22, 27	negsubdi2d 8303	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
29	23, 25	readdcld 8006	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
30	23, 25	resubcld 8357	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
31	26	2timesd 9180	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
32	24, 26, 26	pnncand 8326	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	31, 32	eqtr4d 2225	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34		2rp 9677	. . . . . . . . . . . 12
35	34	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
36		simprr 531	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37	25, 20, 35, 36	ltmul2dd 9772	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
38	33, 37	eqbrtrrd 4042	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39	29, 30, 21, 38	ltsub23d 8526	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	28, 39	eqbrtrd 4040	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	24, 26, 24	nppcan3d 8314	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
42	24	2timesd 9180	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
43	41, 42	eqtr4d 2225	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
44		simprl 529	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
45	23, 20, 35, 44	ltmul2dd 9772	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
46	43, 45	eqbrtrd 4040	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
47	30, 29, 21	ltaddsubd 8521	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
48	46, 47	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
49	40, 48	jca 306	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
50	21, 29	resubcld 8357	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
51	30, 50	absltd 11202	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
52	49, 51	mpbird 167	. . . . 5 $/\$ $/\$ $/\$ $/\$
53	30	recnd 8005	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54	53	abscld 11209	. . . . . 6 $/\$ $/\$ $/\$ $/\$
55	29, 54, 21	ltaddsub2d 8522	. . . . 5 $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
57	29, 54	readdcld 8006	. . . . 5 $/\$ $/\$ $/\$ $/\$
58	57, 20, 35	ltdivmuld 9767	. . . 4 $/\$ $/\$ $/\$ $/\$
59	56, 58	mpbird 167	. . 3 $/\$ $/\$ $/\$ $/\$
60	17, 59	eqbrtrd 4040	. 2 $/\$ $/\$ $/\$ $/\$
61	14, 60	impbida 596	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 980

wceq 1364

wcel 2160

cpr 3608 class class class wbr 4018

cfv 5231 (class class class)co 5891

csup 7000

cr 7829

caddc 7833

cmul 7835

clt 8011

cle 8012

cmin 8147

cneg 8148

cdiv 8648

c2 8989

crp 9672

cabs 11025

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7921 ax-resscn 7922 ax-1cn 7923 ax-1re 7924 ax-icn 7925 ax-addcl 7926 ax-addrcl 7927 ax-mulcl 7928 ax-mulrcl 7929 ax-addcom 7930 ax-mulcom 7931 ax-addass 7932 ax-mulass 7933 ax-distr 7934 ax-i2m1 7935 ax-0lt1 7936 ax-1rid 7937 ax-0id 7938 ax-rnegex 7939 ax-precex 7940 ax-cnre 7941 ax-pre-ltirr 7942 ax-pre-ltwlin 7943 ax-pre-lttrn 7944 ax-pre-apti 7945 ax-pre-ltadd 7946 ax-pre-mulgt0 7947 ax-pre-mulext 7948 ax-arch 7949 ax-caucvg 7950

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 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-sup 7002 df-pnf 8013 df-mnf 8014 df-xr 8015 df-ltxr 8016 df-le 8017 df-sub 8149 df-neg 8150 df-reap 8551 df-ap 8558 df-div 8649 df-inn 8939 df-2 8997 df-3 8998 df-4 8999 df-n0 9196 df-z 9273 df-uz 9548 df-rp 9673 df-seqfrec 10465 df-exp 10539 df-cj 10870 df-re 10871 df-im 10872 df-rsqrt 11026 df-abs 11027

This theorem is referenced by: ltmininf 11262 xrmaxltsup 11285 suplociccreex 14505

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