maxltsup - Intuitionistic Logic Explorer

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Theorem maxltsup 11984

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 1031	. . . 4 $/\$ $/\$ $/\$
2		simpl2 1032	. . . . 5 $/\$ $/\$ $/\$
3		maxcl 11976	. . . . 5 $/\$
4	1, 2, 3	syl2anc 415	. . . 4 $/\$ $/\$ $/\$
5		simpl3 1033	. . . 4 $/\$ $/\$ $/\$
6		maxle1 11977	. . . . . 6 $/\$
7	6	3adant3 1048	. . . . 5 $/\$ $/\$
8	7	adantr 276	. . . 4 $/\$ $/\$ $/\$
9		simpr 110	. . . 4 $/\$ $/\$ $/\$
10	1, 4, 5, 8, 9	lelttrd 8451	. . 3 $/\$ $/\$ $/\$
11		maxle2 11978	. . . . 5 $/\$
12	1, 2, 11	syl2anc 415	. . . 4 $/\$ $/\$ $/\$
13	2, 4, 5, 12, 9	lelttrd 8451	. . 3 $/\$ $/\$ $/\$
14	10, 13	jca 306	. 2 $/\$ $/\$ $/\$ $/\$
15		maxabs 11975	. . . . 5 $/\$
16	15	3adant3 1048	. . . 4 $/\$ $/\$
17	16	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$
18		2re 9374	. . . . . . . . . . . 12
19	18	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
20		simpl3 1033	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	19, 20	remulcld 8356	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22	21	recnd 8354	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
23		simpl1 1031	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
24	23	recnd 8354	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
25		simpl2 1032	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
26	25	recnd 8354	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	24, 26	addcld 8345	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	22, 27	negsubdi2d 8653	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
29	23, 25	readdcld 8355	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
30	23, 25	resubcld 8708	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
31	26	2timesd 9548	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
32	24, 26, 26	pnncand 8676	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	31, 32	eqtr4d 2274	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34		2rp 10059	. . . . . . . . . . . 12
35	34	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
36		simprr 537	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37	25, 20, 35, 36	ltmul2dd 10154	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
38	33, 37	eqbrtrrd 4154	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39	29, 30, 21, 38	ltsub23d 8878	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	28, 39	eqbrtrd 4152	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	24, 26, 24	nppcan3d 8664	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
42	24	2timesd 9548	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
43	41, 42	eqtr4d 2274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
44		simprl 535	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
45	23, 20, 35, 44	ltmul2dd 10154	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
46	43, 45	eqbrtrd 4152	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
47	30, 29, 21	ltaddsubd 8873	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
48	46, 47	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
49	40, 48	jca 306	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
50	21, 29	resubcld 8708	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
51	30, 50	absltd 11940	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
52	49, 51	mpbird 167	. . . . 5 $/\$ $/\$ $/\$ $/\$
53	30	recnd 8354	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54	53	abscld 11947	. . . . . 6 $/\$ $/\$ $/\$ $/\$
55	29, 54, 21	ltaddsub2d 8874	. . . . 5 $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
57	29, 54	readdcld 8355	. . . . 5 $/\$ $/\$ $/\$ $/\$
58	57, 20, 35	ltdivmuld 10149	. . . 4 $/\$ $/\$ $/\$ $/\$
59	56, 58	mpbird 167	. . 3 $/\$ $/\$ $/\$ $/\$
60	17, 59	eqbrtrd 4152	. 2 $/\$ $/\$ $/\$ $/\$
61	14, 60	impbida 604	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105 $/\$ w3a 1009

wceq 1402

wcel 2209

cpr 3710 class class class wbr 4130

cfv 5377 (class class class)co 6085

csup 7322

cr 8178

caddc 8182

cmul 8184

clt 8360

cle 8361

cmin 8497

cneg 8498

cdiv 9002

c2 9355

crp 10054

cabs 11763

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4246 ax-sep 4249 ax-nul 4259 ax-pow 4311 ax-pr 4346 ax-un 4578 ax-setind 4684 ax-iinf 4735 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-mulrcl 8278 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-0lt1 8285 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-precex 8289 ax-cnre 8290 ax-pre-ltirr 8291 ax-pre-ltwlin 8292 ax-pre-lttrn 8293 ax-pre-apti 8294 ax-pre-ltadd 8295 ax-pre-mulgt0 8296 ax-pre-mulext 8297 ax-arch 8298 ax-caucvg 8299

This proof depends on definitions: df-bi 117 df-dc 847 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-if 3639 df-pw 3690 df-sn 3715 df-pr 3716 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4131 df-opab 4193 df-mpt 4194 df-tr 4230 df-id 4438 df-po 4441 df-iso 4442 df-iord 4511 df-on 4513 df-ilim 4514 df-suc 4516 df-iom 4738 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-rn 4785 df-res 4786 df-ima 4787 df-iota 5337 df-fun 5379 df-fn 5380 df-f 5381 df-f1 5382 df-fo 5383 df-f1o 5384 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-1st 6374 df-2nd 6375 df-recs 6576 df-frec 6662 df-sup 7324 df-pnf 8362 df-mnf 8363 df-xr 8364 df-ltxr 8365 df-le 8366 df-sub 8499 df-neg 8500 df-reap 8903 df-ap 8910 df-div 9003 df-inn 9305 df-2 9363 df-3 9364 df-4 9365 df-n0 9564 df-z 9645 df-uz 9922 df-rp 10055 df-seqfrec 10885 df-exp 10976 df-cj 11607 df-re 11608 df-im 11609 df-rsqrt 11764 df-abs 11765

This theorem is used by: ltmininf 12001 xrmaxltsup 12024 suplociccreex 15725 hovera 15748

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