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Theorem modqsubdir 10851

Description: Distribute the modulo operation over a subtraction. (Contributed by Jim Kingdon, 26-Oct-2021.)

**Assertion**
Ref	Expression
modqsubdir	$/\$ $/\$ $/\$

**Proof of Theorem modqsubdir**
Step	Hyp	Ref	Expression
1		simpll 531	. . . . 5 $/\$ $/\$ $/\$
2		simprl 535	. . . . 5 $/\$ $/\$ $/\$
3		simprr 537	. . . . 5 $/\$ $/\$ $/\$
4	1, 2, 3	modqcld 10786	. . . 4 $/\$ $/\$ $/\$
5		qre 10041	. . . 4
6	4, 5	syl 14	. . 3 $/\$ $/\$ $/\$
7		simplr 533	. . . . 5 $/\$ $/\$ $/\$
8	7, 2, 3	modqcld 10786	. . . 4 $/\$ $/\$ $/\$
9		qre 10041	. . . 4
10	8, 9	syl 14	. . 3 $/\$ $/\$ $/\$
11	6, 10	subge0d 8870	. 2 $/\$ $/\$ $/\$
12		qsubcl 10054	. . . . . . . 8 $/\$
13	12	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$
14	3	gt0ne0d 8847	. . . . . . . . . 10 $/\$ $/\$ $/\$
15		qdivcl 10059	. . . . . . . . . 10 $/\$ $/\$
16	1, 2, 14, 15	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
17	16	flqcld 10731	. . . . . . . 8 $/\$ $/\$ $/\$
18		qdivcl 10059	. . . . . . . . . 10 $/\$ $/\$
19	7, 2, 14, 18	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
20	19	flqcld 10731	. . . . . . . 8 $/\$ $/\$ $/\$
21	17, 20	zsubcld 9784	. . . . . . 7 $/\$ $/\$ $/\$
22		modqcyc2 10818	. . . . . . 7 $/\$ $/\$ $/\$
23	13, 21, 2, 3, 22	syl22anc 1279	. . . . . 6 $/\$ $/\$ $/\$
24		qcn 10050	. . . . . . . . . 10
25	1, 24	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$
26		qcn 10050	. . . . . . . . . 10
27	7, 26	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$
28		zq 10042	. . . . . . . . . . . 12
29	17, 28	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$
30		qmulcl 10053	. . . . . . . . . . 11 $/\$
31	2, 29, 30	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$ $/\$
32		qcn 10050	. . . . . . . . . 10
33	31, 32	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$
34		zq 10042	. . . . . . . . . . . 12
35	20, 34	syl 14	. . . . . . . . . . 11 $/\$ $/\$ $/\$
36		qmulcl 10053	. . . . . . . . . . 11 $/\$
37	2, 35, 36	syl2anc 415	. . . . . . . . . 10 $/\$ $/\$ $/\$
38		qcn 10050	. . . . . . . . . 10
39	37, 38	syl 14	. . . . . . . . 9 $/\$ $/\$ $/\$
40	25, 27, 33, 39	sub4d 8693	. . . . . . . 8 $/\$ $/\$ $/\$
41		qcn 10050	. . . . . . . . . . 11
42	2, 41	syl 14	. . . . . . . . . 10 $/\$ $/\$ $/\$
43	17	zcnd 9780	. . . . . . . . . 10 $/\$ $/\$ $/\$
44	20	zcnd 9780	. . . . . . . . . 10 $/\$ $/\$ $/\$
45	42, 43, 44	subdid 8748	. . . . . . . . 9 $/\$ $/\$ $/\$
46	45	oveq2d 6104	. . . . . . . 8 $/\$ $/\$ $/\$
47		modqval 10782	. . . . . . . . . 10 $/\$ $/\$
48	1, 2, 3, 47	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
49		modqval 10782	. . . . . . . . . 10 $/\$ $/\$
50	7, 2, 3, 49	syl3anc 1278	. . . . . . . . 9 $/\$ $/\$ $/\$
51	48, 50	oveq12d 6106	. . . . . . . 8 $/\$ $/\$ $/\$
52	40, 46, 51	3eqtr4d 2281	. . . . . . 7 $/\$ $/\$ $/\$
53	52	oveq1d 6103	. . . . . 6 $/\$ $/\$ $/\$
54	23, 53	eqtr3d 2273	. . . . 5 $/\$ $/\$ $/\$
55	54	adantr 276	. . . 4 $/\$ $/\$ $/\$ $/\$
56		qsubcl 10054	. . . . . . 7 $/\$
57	4, 8, 56	syl2anc 415	. . . . . 6 $/\$ $/\$ $/\$
58	57	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
59	2	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
60		simpr 110	. . . . 5 $/\$ $/\$ $/\$ $/\$
61	6, 10	resubcld 8715	. . . . . . 7 $/\$ $/\$ $/\$
62		qre 10041	. . . . . . . 8
63	2, 62	syl 14	. . . . . . 7 $/\$ $/\$ $/\$
64		modqge0 10790	. . . . . . . . 9 $/\$ $/\$
65	7, 2, 3, 64	syl3anc 1278	. . . . . . . 8 $/\$ $/\$ $/\$
66	6, 10	subge02d 8872	. . . . . . . 8 $/\$ $/\$ $/\$
67	65, 66	mpbid 147	. . . . . . 7 $/\$ $/\$ $/\$
68		modqlt 10791	. . . . . . . 8 $/\$ $/\$
69	1, 2, 3, 68	syl3anc 1278	. . . . . . 7 $/\$ $/\$ $/\$
70	61, 6, 63, 67, 69	lelttrd 8458	. . . . . 6 $/\$ $/\$ $/\$
71	70	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$
72		modqid 10807	. . . . 5 $/\$ $/\$ $/\$
73	58, 59, 60, 71, 72	syl22anc 1279	. . . 4 $/\$ $/\$ $/\$ $/\$
74	55, 73	eqtrd 2271	. . 3 $/\$ $/\$ $/\$ $/\$
75		modqge0 10790	. . . . . 6 $/\$ $/\$
76	13, 2, 3, 75	syl3anc 1278	. . . . 5 $/\$ $/\$ $/\$
77	76	adantr 276	. . . 4 $/\$ $/\$ $/\$ $/\$
78		simpr 110	. . . 4 $/\$ $/\$ $/\$ $/\$
79	77, 78	breqtrd 4157	. . 3 $/\$ $/\$ $/\$ $/\$
80	74, 79	impbida 604	. 2 $/\$ $/\$ $/\$
81	11, 80	bitr3d 190	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

This proof depends on syntax axioms:

wi 4 $/\$ wa 104

wb 105

wceq 1402

wcel 2209

wne 2420 class class class wbr 4131

cfv 5380 (class class class)co 6088

cc 8183

cr 8184

cc0 8185

cmul 8190

clt 8366

cle 8367

cmin 8504

cdiv 9010

cz 9655

cq 10035

cfl 10720

cmo 10780

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-sep 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 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

This proof depends on definitions: df-bi 117 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-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-id 4440 df-po 4443 df-iso 4444 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-fv 5388 df-riota 6041 df-ov 6091 df-oprab 6092 df-mpo 6093 df-1st 6378 df-2nd 6379 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-n0 9575 df-z 9656 df-q 10036 df-rp 10072 df-fl 10722 df-mod 10781

This theorem is used by: modqeqmodmin 10852 4sqlem12 13210

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