maxltsup - Intuitionistic Logic Explorer

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Theorem maxltsup 11022

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 985	. . . 4 $/\$ $/\$ $/\$
2		simpl2 986	. . . . 5 $/\$ $/\$ $/\$
3		maxcl 11014	. . . . 5 $/\$
4	1, 2, 3	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
5		simpl3 987	. . . 4 $/\$ $/\$ $/\$
6		maxle1 11015	. . . . . 6 $/\$
7	6	3adant3 1002	. . . . 5 $/\$ $/\$
8	7	adantr 274	. . . 4 $/\$ $/\$ $/\$
9		simpr 109	. . . 4 $/\$ $/\$ $/\$
10	1, 4, 5, 8, 9	lelttrd 7911	. . 3 $/\$ $/\$ $/\$
11		maxle2 11016	. . . . 5 $/\$
12	1, 2, 11	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
13	2, 4, 5, 12, 9	lelttrd 7911	. . 3 $/\$ $/\$ $/\$
14	10, 13	jca 304	. 2 $/\$ $/\$ $/\$ $/\$
15		maxabs 11013	. . . . 5 $/\$
16	15	3adant3 1002	. . . 4 $/\$ $/\$
17	16	adantr 274	. . 3 $/\$ $/\$ $/\$ $/\$
18		2re 8814	. . . . . . . . . . . 12
19	18	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
20		simpl3 987	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	19, 20	remulcld 7820	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22	21	recnd 7818	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
23		simpl1 985	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
24	23	recnd 7818	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
25		simpl2 986	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
26	25	recnd 7818	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	24, 26	addcld 7809	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	22, 27	negsubdi2d 8113	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
29	23, 25	readdcld 7819	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
30	23, 25	resubcld 8167	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
31	26	2timesd 8986	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
32	24, 26, 26	pnncand 8136	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	31, 32	eqtr4d 2176	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34		2rp 9475	. . . . . . . . . . . 12
35	34	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
36		simprr 522	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37	25, 20, 35, 36	ltmul2dd 9570	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
38	33, 37	eqbrtrrd 3960	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39	29, 30, 21, 38	ltsub23d 8336	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	28, 39	eqbrtrd 3958	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	24, 26, 24	nppcan3d 8124	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
42	24	2timesd 8986	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
43	41, 42	eqtr4d 2176	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
44		simprl 521	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
45	23, 20, 35, 44	ltmul2dd 9570	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
46	43, 45	eqbrtrd 3958	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
47	30, 29, 21	ltaddsubd 8331	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
48	46, 47	mpbid 146	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
49	40, 48	jca 304	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
50	21, 29	resubcld 8167	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
51	30, 50	absltd 10978	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
52	49, 51	mpbird 166	. . . . 5 $/\$ $/\$ $/\$ $/\$
53	30	recnd 7818	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54	53	abscld 10985	. . . . . 6 $/\$ $/\$ $/\$ $/\$
55	29, 54, 21	ltaddsub2d 8332	. . . . 5 $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbird 166	. . . 4 $/\$ $/\$ $/\$ $/\$
57	29, 54	readdcld 7819	. . . . 5 $/\$ $/\$ $/\$ $/\$
58	57, 20, 35	ltdivmuld 9565	. . . 4 $/\$ $/\$ $/\$ $/\$
59	56, 58	mpbird 166	. . 3 $/\$ $/\$ $/\$ $/\$
60	17, 59	eqbrtrd 3958	. 2 $/\$ $/\$ $/\$ $/\$
61	14, 60	impbida 586	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 963

wceq 1332

wcel 1481

cpr 3533 class class class wbr 3937

cfv 5131 (class class class)co 5782

csup 6877

cr 7643

caddc 7647

cmul 7649

clt 7824

cle 7825

cmin 7957

cneg 7958

cdiv 8456

c2 8795

crp 9470

cabs 10801

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 ax-pre-mulext 7762 ax-arch 7763 ax-caucvg 7764

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rmo 2425 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-if 3480 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-tr 4035 df-id 4223 df-po 4226 df-iso 4227 df-iord 4296 df-on 4298 df-ilim 4299 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-1st 6046 df-2nd 6047 df-recs 6210 df-frec 6296 df-sup 6879 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 df-div 8457 df-inn 8745 df-2 8803 df-3 8804 df-4 8805 df-n0 9002 df-z 9079 df-uz 9351 df-rp 9471 df-seqfrec 10250 df-exp 10324 df-cj 10646 df-re 10647 df-im 10648 df-rsqrt 10802 df-abs 10803

This theorem is referenced by: ltmininf 11038 xrmaxltsup 11059 suplociccreex 12810

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