maxltsup - Intuitionistic Logic Explorer

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Theorem maxltsup 11400

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 11392	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
5		simpl3 1004	. . . 4 $/\$ $/\$ $/\$
6		maxle1 11393	. . . . . 6 $/\$
7	6	3adant3 1019	. . . . 5 $/\$ $/\$
8	7	adantr 276	. . . 4 $/\$ $/\$ $/\$
9		simpr 110	. . . 4 $/\$ $/\$ $/\$
10	1, 4, 5, 8, 9	lelttrd 8168	. . 3 $/\$ $/\$ $/\$
11		maxle2 11394	. . . . 5 $/\$
12	1, 2, 11	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
13	2, 4, 5, 12, 9	lelttrd 8168	. . 3 $/\$ $/\$ $/\$
14	10, 13	jca 306	. 2 $/\$ $/\$ $/\$ $/\$
15		maxabs 11391	. . . . 5 $/\$
16	15	3adant3 1019	. . . 4 $/\$ $/\$
17	16	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$
18		2re 9077	. . . . . . . . . . . 12
19	18	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
20		simpl3 1004	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	19, 20	remulcld 8074	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22	21	recnd 8072	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
23		simpl1 1002	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
24	23	recnd 8072	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
25		simpl2 1003	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
26	25	recnd 8072	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	24, 26	addcld 8063	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	22, 27	negsubdi2d 8370	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
29	23, 25	readdcld 8073	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
30	23, 25	resubcld 8424	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
31	26	2timesd 9251	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
32	24, 26, 26	pnncand 8393	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	31, 32	eqtr4d 2232	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34		2rp 9750	. . . . . . . . . . . 12
35	34	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
36		simprr 531	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37	25, 20, 35, 36	ltmul2dd 9845	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
38	33, 37	eqbrtrrd 4058	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39	29, 30, 21, 38	ltsub23d 8594	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	28, 39	eqbrtrd 4056	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	24, 26, 24	nppcan3d 8381	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
42	24	2timesd 9251	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
43	41, 42	eqtr4d 2232	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
44		simprl 529	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
45	23, 20, 35, 44	ltmul2dd 9845	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
46	43, 45	eqbrtrd 4056	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
47	30, 29, 21	ltaddsubd 8589	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
48	46, 47	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
49	40, 48	jca 306	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
50	21, 29	resubcld 8424	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
51	30, 50	absltd 11356	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
52	49, 51	mpbird 167	. . . . 5 $/\$ $/\$ $/\$ $/\$
53	30	recnd 8072	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54	53	abscld 11363	. . . . . 6 $/\$ $/\$ $/\$ $/\$
55	29, 54, 21	ltaddsub2d 8590	. . . . 5 $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
57	29, 54	readdcld 8073	. . . . 5 $/\$ $/\$ $/\$ $/\$
58	57, 20, 35	ltdivmuld 9840	. . . 4 $/\$ $/\$ $/\$ $/\$
59	56, 58	mpbird 167	. . 3 $/\$ $/\$ $/\$ $/\$
60	17, 59	eqbrtrd 4056	. 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 2167

cpr 3624 class class class wbr 4034

cfv 5259 (class class class)co 5925

csup 7057

cr 7895

caddc 7899

cmul 7901

clt 8078

cle 8079

cmin 8214

cneg 8215

cdiv 8716

c2 9058

crp 9745

cabs 11179

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4149 ax-sep 4152 ax-nul 4160 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-iinf 4625 ax-cnex 7987 ax-resscn 7988 ax-1cn 7989 ax-1re 7990 ax-icn 7991 ax-addcl 7992 ax-addrcl 7993 ax-mulcl 7994 ax-mulrcl 7995 ax-addcom 7996 ax-mulcom 7997 ax-addass 7998 ax-mulass 7999 ax-distr 8000 ax-i2m1 8001 ax-0lt1 8002 ax-1rid 8003 ax-0id 8004 ax-rnegex 8005 ax-precex 8006 ax-cnre 8007 ax-pre-ltirr 8008 ax-pre-ltwlin 8009 ax-pre-lttrn 8010 ax-pre-apti 8011 ax-pre-ltadd 8012 ax-pre-mulgt0 8013 ax-pre-mulext 8014 ax-arch 8015 ax-caucvg 8016

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 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3452 df-if 3563 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-int 3876 df-iun 3919 df-br 4035 df-opab 4096 df-mpt 4097 df-tr 4133 df-id 4329 df-po 4332 df-iso 4333 df-iord 4402 df-on 4404 df-ilim 4405 df-suc 4407 df-iom 4628 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-rn 4675 df-res 4676 df-ima 4677 df-iota 5220 df-fun 5261 df-fn 5262 df-f 5263 df-f1 5264 df-fo 5265 df-f1o 5266 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-1st 6207 df-2nd 6208 df-recs 6372 df-frec 6458 df-sup 7059 df-pnf 8080 df-mnf 8081 df-xr 8082 df-ltxr 8083 df-le 8084 df-sub 8216 df-neg 8217 df-reap 8619 df-ap 8626 df-div 8717 df-inn 9008 df-2 9066 df-3 9067 df-4 9068 df-n0 9267 df-z 9344 df-uz 9619 df-rp 9746 df-seqfrec 10557 df-exp 10648 df-cj 11024 df-re 11025 df-im 11026 df-rsqrt 11180 df-abs 11181

This theorem is referenced by: ltmininf 11417 xrmaxltsup 11440 suplociccreex 14944 hovera 14967

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