remullem - Intuitionistic Logic Explorer

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Theorem remullem 10484

Description: Lemma for remul 10485, immul 10492, and cjmul 10498. (Contributed by NM, 28-Jul-1999.) (Revised by Mario Carneiro, 14-Jul-2014.)

**Assertion**
Ref	Expression
remullem	$/\$ $/\$ $/\$

**Proof of Theorem remullem**
Step	Hyp	Ref	Expression
1		replim 10472	. . . . . 6
2		replim 10472	. . . . . 6
3	1, 2	oveqan12d 5725	. . . . 5 $/\$
4		recl 10466	. . . . . . . . 9
5	4	adantr 272	. . . . . . . 8 $/\$
6	5	recnd 7666	. . . . . . 7 $/\$
7		ax-icn 7590	. . . . . . . 8
8		imcl 10467	. . . . . . . . . 10
9	8	adantr 272	. . . . . . . . 9 $/\$
10	9	recnd 7666	. . . . . . . 8 $/\$
11		mulcl 7619	. . . . . . . 8 $/\$
12	7, 10, 11	sylancr 408	. . . . . . 7 $/\$
13	6, 12	addcld 7657	. . . . . 6 $/\$
14		recl 10466	. . . . . . . 8
15	14	adantl 273	. . . . . . 7 $/\$
16	15	recnd 7666	. . . . . 6 $/\$
17		imcl 10467	. . . . . . . . 9
18	17	adantl 273	. . . . . . . 8 $/\$
19	18	recnd 7666	. . . . . . 7 $/\$
20		mulcl 7619	. . . . . . 7 $/\$
21	7, 19, 20	sylancr 408	. . . . . 6 $/\$
22	13, 16, 21	adddid 7662	. . . . 5 $/\$
23	6, 12, 16	adddird 7663	. . . . . . 7 $/\$
24	6, 12, 21	adddird 7663	. . . . . . 7 $/\$
25	23, 24	oveq12d 5724	. . . . . 6 $/\$
26	5, 15	remulcld 7668	. . . . . . . 8 $/\$
27	26	recnd 7666	. . . . . . 7 $/\$
28	12, 21	mulcld 7658	. . . . . . 7 $/\$
29	12, 16	mulcld 7658	. . . . . . 7 $/\$
30	6, 21	mulcld 7658	. . . . . . 7 $/\$
31	27, 28, 29, 30	add42d 7803	. . . . . 6 $/\$
32	7	a1i 9	. . . . . . . . . . 11 $/\$
33	32, 10, 32, 19	mul4d 7788	. . . . . . . . . 10 $/\$
34		ixi 8211	. . . . . . . . . . . 12
35	34	oveq1i 5716	. . . . . . . . . . 11
36	9, 18	remulcld 7668	. . . . . . . . . . . . 13 $/\$
37	36	recnd 7666	. . . . . . . . . . . 12 $/\$
38	37	mulm1d 8039	. . . . . . . . . . 11 $/\$
39	35, 38	syl5eq 2144	. . . . . . . . . 10 $/\$
40	33, 39	eqtrd 2132	. . . . . . . . 9 $/\$
41	40	oveq2d 5722	. . . . . . . 8 $/\$
42	27, 37	negsubd 7950	. . . . . . . 8 $/\$
43	41, 42	eqtrd 2132	. . . . . . 7 $/\$
44	9, 15	remulcld 7668	. . . . . . . . . . 11 $/\$
45	44	recnd 7666	. . . . . . . . . 10 $/\$
46		mulcl 7619	. . . . . . . . . 10 $/\$
47	7, 45, 46	sylancr 408	. . . . . . . . 9 $/\$
48	5, 18	remulcld 7668	. . . . . . . . . . 11 $/\$
49	48	recnd 7666	. . . . . . . . . 10 $/\$
50		mulcl 7619	. . . . . . . . . 10 $/\$
51	7, 49, 50	sylancr 408	. . . . . . . . 9 $/\$
52	47, 51	addcomd 7784	. . . . . . . 8 $/\$
53	32, 10, 16	mulassd 7661	. . . . . . . . 9 $/\$
54	6, 32, 19	mul12d 7785	. . . . . . . . 9 $/\$
55	53, 54	oveq12d 5724	. . . . . . . 8 $/\$
56	32, 49, 45	adddid 7662	. . . . . . . 8 $/\$
57	52, 55, 56	3eqtr4d 2142	. . . . . . 7 $/\$
58	43, 57	oveq12d 5724	. . . . . 6 $/\$
59	25, 31, 58	3eqtr2d 2138	. . . . 5 $/\$
60	3, 22, 59	3eqtrd 2136	. . . 4 $/\$
61	60	fveq2d 5357	. . 3 $/\$
62	26, 36	resubcld 8010	. . . 4 $/\$
63	48, 44	readdcld 7667	. . . 4 $/\$
64		crre 10470	. . . 4 $/\$
65	62, 63, 64	syl2anc 406	. . 3 $/\$
66	61, 65	eqtrd 2132	. 2 $/\$
67	60	fveq2d 5357	. . 3 $/\$
68		crim 10471	. . . 4 $/\$
69	62, 63, 68	syl2anc 406	. . 3 $/\$
70	67, 69	eqtrd 2132	. 2 $/\$
71		mulcl 7619	. . . 4 $/\$
72		remim 10473	. . . 4
73	71, 72	syl 14	. . 3 $/\$
74		remim 10473	. . . . 5
75		remim 10473	. . . . 5
76	74, 75	oveqan12d 5725	. . . 4 $/\$
77	16, 21	subcld 7944	. . . . 5 $/\$
78	6, 12, 77	subdird 8044	. . . 4 $/\$
79	27, 30, 29, 28	subadd4d 7992	. . . . 5 $/\$
80	6, 16, 21	subdid 8043	. . . . . 6 $/\$
81	12, 16, 21	subdid 8043	. . . . . 6 $/\$
82	80, 81	oveq12d 5724	. . . . 5 $/\$
83	65, 61, 43	3eqtr4d 2142	. . . . . 6 $/\$
84	70	oveq2d 5722	. . . . . . 7 $/\$
85	54, 53	oveq12d 5724	. . . . . . 7 $/\$
86	56, 84, 85	3eqtr4d 2142	. . . . . 6 $/\$
87	83, 86	oveq12d 5724	. . . . 5 $/\$
88	79, 82, 87	3eqtr4d 2142	. . . 4 $/\$
89	76, 78, 88	3eqtrd 2136	. . 3 $/\$
90	73, 89	eqtr4d 2135	. 2 $/\$
91	66, 70, 90	3jca 1129	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 930

wceq 1299

wcel 1448

cfv 5059 (class class class)co 5706

cc 7498

cr 7499

c1 7501

ci 7502

caddc 7503

cmul 7505

cmin 7804

cneg 7805

ccj 10452

cre 10453

cim 10454

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-1cn 7588 ax-1re 7589 ax-icn 7590 ax-addcl 7591 ax-addrcl 7592 ax-mulcl 7593 ax-mulrcl 7594 ax-addcom 7595 ax-mulcom 7596 ax-addass 7597 ax-mulass 7598 ax-distr 7599 ax-i2m1 7600 ax-0lt1 7601 ax-1rid 7602 ax-0id 7603 ax-rnegex 7604 ax-precex 7605 ax-cnre 7606 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 ax-pre-ltadd 7611 ax-pre-mulgt0 7612 ax-pre-mulext 7613

This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-mpt 3931 df-id 4153 df-po 4156 df-iso 4157 df-xp 4483 df-rel 4484 df-cnv 4485 df-co 4486 df-dm 4487 df-rn 4488 df-res 4489 df-ima 4490 df-iota 5024 df-fun 5061 df-fn 5062 df-f 5063 df-fv 5067 df-riota 5662 df-ov 5709 df-oprab 5710 df-mpo 5711 df-pnf 7674 df-mnf 7675 df-xr 7676 df-ltxr 7677 df-le 7678 df-sub 7806 df-neg 7807 df-reap 8203 df-ap 8210 df-div 8294 df-2 8637 df-cj 10455 df-re 10456 df-im 10457

This theorem is referenced by: remul 10485 immul 10492 cjmul 10498

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