maxltsup - Intuitionistic Logic Explorer

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Theorem maxltsup 11604

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 1003	. . . 4 $/\$ $/\$ $/\$
2		simpl2 1004	. . . . 5 $/\$ $/\$ $/\$
3		maxcl 11596	. . . . 5 $/\$
4	1, 2, 3	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
5		simpl3 1005	. . . 4 $/\$ $/\$ $/\$
6		maxle1 11597	. . . . . 6 $/\$
7	6	3adant3 1020	. . . . 5 $/\$ $/\$
8	7	adantr 276	. . . 4 $/\$ $/\$ $/\$
9		simpr 110	. . . 4 $/\$ $/\$ $/\$
10	1, 4, 5, 8, 9	lelttrd 8217	. . 3 $/\$ $/\$ $/\$
11		maxle2 11598	. . . . 5 $/\$
12	1, 2, 11	syl2anc 411	. . . 4 $/\$ $/\$ $/\$
13	2, 4, 5, 12, 9	lelttrd 8217	. . 3 $/\$ $/\$ $/\$
14	10, 13	jca 306	. 2 $/\$ $/\$ $/\$ $/\$
15		maxabs 11595	. . . . 5 $/\$
16	15	3adant3 1020	. . . 4 $/\$ $/\$
17	16	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$
18		2re 9126	. . . . . . . . . . . 12
19	18	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
20		simpl3 1005	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	19, 20	remulcld 8123	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22	21	recnd 8121	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
23		simpl1 1003	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
24	23	recnd 8121	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
25		simpl2 1004	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
26	25	recnd 8121	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	24, 26	addcld 8112	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	22, 27	negsubdi2d 8419	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
29	23, 25	readdcld 8122	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
30	23, 25	resubcld 8473	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
31	26	2timesd 9300	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
32	24, 26, 26	pnncand 8442	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	31, 32	eqtr4d 2242	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34		2rp 9800	. . . . . . . . . . . 12
35	34	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
36		simprr 531	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37	25, 20, 35, 36	ltmul2dd 9895	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
38	33, 37	eqbrtrrd 4075	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39	29, 30, 21, 38	ltsub23d 8643	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	28, 39	eqbrtrd 4073	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	24, 26, 24	nppcan3d 8430	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
42	24	2timesd 9300	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
43	41, 42	eqtr4d 2242	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
44		simprl 529	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
45	23, 20, 35, 44	ltmul2dd 9895	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
46	43, 45	eqbrtrd 4073	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
47	30, 29, 21	ltaddsubd 8638	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
48	46, 47	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
49	40, 48	jca 306	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
50	21, 29	resubcld 8473	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
51	30, 50	absltd 11560	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
52	49, 51	mpbird 167	. . . . 5 $/\$ $/\$ $/\$ $/\$
53	30	recnd 8121	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54	53	abscld 11567	. . . . . 6 $/\$ $/\$ $/\$ $/\$
55	29, 54, 21	ltaddsub2d 8639	. . . . 5 $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
57	29, 54	readdcld 8122	. . . . 5 $/\$ $/\$ $/\$ $/\$
58	57, 20, 35	ltdivmuld 9890	. . . 4 $/\$ $/\$ $/\$ $/\$
59	56, 58	mpbird 167	. . 3 $/\$ $/\$ $/\$ $/\$
60	17, 59	eqbrtrd 4073	. 2 $/\$ $/\$ $/\$ $/\$
61	14, 60	impbida 596	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 981

wceq 1373

wcel 2177

cpr 3639 class class class wbr 4051

cfv 5280 (class class class)co 5957

csup 7099

cr 7944

caddc 7948

cmul 7950

clt 8127

cle 8128

cmin 8263

cneg 8264

cdiv 8765

c2 9107

crp 9795

cabs 11383

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-coll 4167 ax-sep 4170 ax-nul 4178 ax-pow 4226 ax-pr 4261 ax-un 4488 ax-setind 4593 ax-iinf 4644 ax-cnex 8036 ax-resscn 8037 ax-1cn 8038 ax-1re 8039 ax-icn 8040 ax-addcl 8041 ax-addrcl 8042 ax-mulcl 8043 ax-mulrcl 8044 ax-addcom 8045 ax-mulcom 8046 ax-addass 8047 ax-mulass 8048 ax-distr 8049 ax-i2m1 8050 ax-0lt1 8051 ax-1rid 8052 ax-0id 8053 ax-rnegex 8054 ax-precex 8055 ax-cnre 8056 ax-pre-ltirr 8057 ax-pre-ltwlin 8058 ax-pre-lttrn 8059 ax-pre-apti 8060 ax-pre-ltadd 8061 ax-pre-mulgt0 8062 ax-pre-mulext 8063 ax-arch 8064 ax-caucvg 8065

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-eu 2058 df-mo 2059 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-ral 2490 df-rex 2491 df-reu 2492 df-rmo 2493 df-rab 2494 df-v 2775 df-sbc 3003 df-csb 3098 df-dif 3172 df-un 3174 df-in 3176 df-ss 3183 df-nul 3465 df-if 3576 df-pw 3623 df-sn 3644 df-pr 3645 df-op 3647 df-uni 3857 df-int 3892 df-iun 3935 df-br 4052 df-opab 4114 df-mpt 4115 df-tr 4151 df-id 4348 df-po 4351 df-iso 4352 df-iord 4421 df-on 4423 df-ilim 4424 df-suc 4426 df-iom 4647 df-xp 4689 df-rel 4690 df-cnv 4691 df-co 4692 df-dm 4693 df-rn 4694 df-res 4695 df-ima 4696 df-iota 5241 df-fun 5282 df-fn 5283 df-f 5284 df-f1 5285 df-fo 5286 df-f1o 5287 df-fv 5288 df-riota 5912 df-ov 5960 df-oprab 5961 df-mpo 5962 df-1st 6239 df-2nd 6240 df-recs 6404 df-frec 6490 df-sup 7101 df-pnf 8129 df-mnf 8130 df-xr 8131 df-ltxr 8132 df-le 8133 df-sub 8265 df-neg 8266 df-reap 8668 df-ap 8675 df-div 8766 df-inn 9057 df-2 9115 df-3 9116 df-4 9117 df-n0 9316 df-z 9393 df-uz 9669 df-rp 9796 df-seqfrec 10615 df-exp 10706 df-cj 11228 df-re 11229 df-im 11230 df-rsqrt 11384 df-abs 11385

This theorem is referenced by: ltmininf 11621 xrmaxltsup 11644 suplociccreex 15171 hovera 15194

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