maxltsup - Intuitionistic Logic Explorer

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Theorem maxltsup 11118

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 11110	. . . . 5 $/\$
4	1, 2, 3	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
5		simpl3 987	. . . 4 $/\$ $/\$ $/\$
6		maxle1 11111	. . . . . 6 $/\$
7	6	3adant3 1002	. . . . 5 $/\$ $/\$
8	7	adantr 274	. . . 4 $/\$ $/\$ $/\$
9		simpr 109	. . . 4 $/\$ $/\$ $/\$
10	1, 4, 5, 8, 9	lelttrd 8000	. . 3 $/\$ $/\$ $/\$
11		maxle2 11112	. . . . 5 $/\$
12	1, 2, 11	syl2anc 409	. . . 4 $/\$ $/\$ $/\$
13	2, 4, 5, 12, 9	lelttrd 8000	. . 3 $/\$ $/\$ $/\$
14	10, 13	jca 304	. 2 $/\$ $/\$ $/\$ $/\$
15		maxabs 11109	. . . . 5 $/\$
16	15	3adant3 1002	. . . 4 $/\$ $/\$
17	16	adantr 274	. . 3 $/\$ $/\$ $/\$ $/\$
18		2re 8903	. . . . . . . . . . . 12
19	18	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
20		simpl3 987	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	19, 20	remulcld 7908	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22	21	recnd 7906	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
23		simpl1 985	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
24	23	recnd 7906	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
25		simpl2 986	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
26	25	recnd 7906	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	24, 26	addcld 7897	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	22, 27	negsubdi2d 8202	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
29	23, 25	readdcld 7907	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
30	23, 25	resubcld 8256	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
31	26	2timesd 9075	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
32	24, 26, 26	pnncand 8225	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	31, 32	eqtr4d 2193	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34		2rp 9565	. . . . . . . . . . . 12
35	34	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
36		simprr 522	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37	25, 20, 35, 36	ltmul2dd 9660	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
38	33, 37	eqbrtrrd 3988	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39	29, 30, 21, 38	ltsub23d 8425	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	28, 39	eqbrtrd 3986	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	24, 26, 24	nppcan3d 8213	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
42	24	2timesd 9075	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
43	41, 42	eqtr4d 2193	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
44		simprl 521	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
45	23, 20, 35, 44	ltmul2dd 9660	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
46	43, 45	eqbrtrd 3986	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
47	30, 29, 21	ltaddsubd 8420	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
48	46, 47	mpbid 146	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
49	40, 48	jca 304	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
50	21, 29	resubcld 8256	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
51	30, 50	absltd 11074	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
52	49, 51	mpbird 166	. . . . 5 $/\$ $/\$ $/\$ $/\$
53	30	recnd 7906	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54	53	abscld 11081	. . . . . 6 $/\$ $/\$ $/\$ $/\$
55	29, 54, 21	ltaddsub2d 8421	. . . . 5 $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbird 166	. . . 4 $/\$ $/\$ $/\$ $/\$
57	29, 54	readdcld 7907	. . . . 5 $/\$ $/\$ $/\$ $/\$
58	57, 20, 35	ltdivmuld 9655	. . . 4 $/\$ $/\$ $/\$ $/\$
59	56, 58	mpbird 166	. . 3 $/\$ $/\$ $/\$ $/\$
60	17, 59	eqbrtrd 3986	. 2 $/\$ $/\$ $/\$ $/\$
61	14, 60	impbida 586	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103

wb 104 $/\$ w3a 963

wceq 1335

wcel 2128

cpr 3561 class class class wbr 3965

cfv 5170 (class class class)co 5824

csup 6926

cr 7731

caddc 7735

cmul 7737

clt 7912

cle 7913

cmin 8046

cneg 8047

cdiv 8545

c2 8884

crp 9560

cabs 10897

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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 ax-pre-mulext 7850 ax-arch 7851 ax-caucvg 7852

This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-frec 6338 df-sup 6928 df-pnf 7914 df-mnf 7915 df-xr 7916 df-ltxr 7917 df-le 7918 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 df-div 8546 df-inn 8834 df-2 8892 df-3 8893 df-4 8894 df-n0 9091 df-z 9168 df-uz 9440 df-rp 9561 df-seqfrec 10345 df-exp 10419 df-cj 10742 df-re 10743 df-im 10744 df-rsqrt 10898 df-abs 10899

This theorem is referenced by: ltmininf 11134 xrmaxltsup 11155 suplociccreex 13013

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