maxltsup - Intuitionistic Logic Explorer

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Theorem maxltsup 12007

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 11999	. . . . 5 $/\$
4	1, 2, 3	syl2anc 415	. . . 4 $/\$ $/\$ $/\$
5		simpl3 1033	. . . 4 $/\$ $/\$ $/\$
6		maxle1 12000	. . . . . 6 $/\$
7	6	3adant3 1048	. . . . 5 $/\$ $/\$
8	7	adantr 276	. . . 4 $/\$ $/\$ $/\$
9		simpr 110	. . . 4 $/\$ $/\$ $/\$
10	1, 4, 5, 8, 9	lelttrd 8458	. . 3 $/\$ $/\$ $/\$
11		maxle2 12001	. . . . 5 $/\$
12	1, 2, 11	syl2anc 415	. . . 4 $/\$ $/\$ $/\$
13	2, 4, 5, 12, 9	lelttrd 8458	. . 3 $/\$ $/\$ $/\$
14	10, 13	jca 306	. 2 $/\$ $/\$ $/\$ $/\$
15		maxabs 11998	. . . . 5 $/\$
16	15	3adant3 1048	. . . 4 $/\$ $/\$
17	16	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$
18		2re 9383	. . . . . . . . . . . 12
19	18	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
20		simpl3 1033	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
21	19, 20	remulcld 8362	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
22	21	recnd 8360	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
23		simpl1 1031	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
24	23	recnd 8360	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
25		simpl2 1032	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
26	25	recnd 8360	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
27	24, 26	addcld 8351	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
28	22, 27	negsubdi2d 8660	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
29	23, 25	readdcld 8361	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
30	23, 25	resubcld 8715	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
31	26	2timesd 9559	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
32	24, 26, 26	pnncand 8683	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
33	31, 32	eqtr4d 2274	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
34		2rp 10076	. . . . . . . . . . . 12
35	34	a1i 9	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
36		simprr 537	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
37	25, 20, 35, 36	ltmul2dd 10171	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
38	33, 37	eqbrtrrd 4155	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
39	29, 30, 21, 38	ltsub23d 8885	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
40	28, 39	eqbrtrd 4153	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
41	24, 26, 24	nppcan3d 8671	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
42	24	2timesd 9559	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
43	41, 42	eqtr4d 2274	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
44		simprl 535	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$
45	23, 20, 35, 44	ltmul2dd 10171	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$
46	43, 45	eqbrtrd 4153	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
47	30, 29, 21	ltaddsubd 8880	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$
48	46, 47	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
49	40, 48	jca 306	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
50	21, 29	resubcld 8715	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
51	30, 50	absltd 11963	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
52	49, 51	mpbird 167	. . . . 5 $/\$ $/\$ $/\$ $/\$
53	30	recnd 8360	. . . . . . 7 $/\$ $/\$ $/\$ $/\$
54	53	abscld 11970	. . . . . 6 $/\$ $/\$ $/\$ $/\$
55	29, 54, 21	ltaddsub2d 8881	. . . . 5 $/\$ $/\$ $/\$ $/\$
56	52, 55	mpbird 167	. . . 4 $/\$ $/\$ $/\$ $/\$
57	29, 54	readdcld 8361	. . . . 5 $/\$ $/\$ $/\$ $/\$
58	57, 20, 35	ltdivmuld 10166	. . . 4 $/\$ $/\$ $/\$ $/\$
59	56, 58	mpbird 167	. . 3 $/\$ $/\$ $/\$ $/\$
60	17, 59	eqbrtrd 4153	. 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 4131

cfv 5380 (class class class)co 6088

csup 7328

cr 8184

caddc 8188

cmul 8190

clt 8366

cle 8367

cmin 8504

cneg 8505

cdiv 9010

c2 9364

crp 10071

cabs 11785

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 4248 ax-sep 4251 ax-nul 4261 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-iinf 4737 ax-cnex 8276 ax-resscn 8277 ax-1cn 8278 ax-1re 8279 ax-icn 8280 ax-addcl 8281 ax-addrcl 8282 ax-mulcl 8283 ax-mulrcl 8284 ax-addcom 8285 ax-mulcom 8286 ax-addass 8287 ax-mulass 8288 ax-distr 8289 ax-i2m1 8290 ax-0lt1 8291 ax-1rid 8292 ax-0id 8293 ax-rnegex 8294 ax-precex 8295 ax-cnre 8296 ax-pre-ltirr 8297 ax-pre-ltwlin 8298 ax-pre-lttrn 8299 ax-pre-apti 8300 ax-pre-ltadd 8301 ax-pre-mulgt0 8302 ax-pre-mulext 8303 ax-arch 8304 ax-caucvg 8305

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 4132 df-opab 4194 df-mpt 4195 df-tr 4232 df-id 4440 df-po 4443 df-iso 4444 df-iord 4513 df-on 4515 df-ilim 4516 df-suc 4518 df-iom 4740 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 df-recs 6580 df-frec 6666 df-sup 7330 df-pnf 8368 df-mnf 8369 df-xr 8370 df-ltxr 8371 df-le 8372 df-sub 8506 df-neg 8507 df-reap 8911 df-ap 8918 df-div 9011 df-inn 9314 df-2 9372 df-3 9373 df-4 9374 df-n0 9575 df-z 9656 df-uz 9938 df-rp 10072 df-seqfrec 10906 df-exp 10997 df-cj 11629 df-re 11630 df-im 11631 df-rsqrt 11786 df-abs 11787

This theorem is used by: ltmininf 12025 xrmaxltsup 12049 suplociccreex 15850 hovera 15873

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