remullem - Intuitionistic Logic Explorer

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Theorem remullem 10643

Description: Lemma for remul 10644, immul 10651, and cjmul 10657. (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 10631	. . . . . 6
2		replim 10631	. . . . . 6
3	1, 2	oveqan12d 5793	. . . . 5 $/\$
4		recl 10625	. . . . . . . . 9
5	4	adantr 274	. . . . . . . 8 $/\$
6	5	recnd 7794	. . . . . . 7 $/\$
7		ax-icn 7715	. . . . . . . 8
8		imcl 10626	. . . . . . . . . 10
9	8	adantr 274	. . . . . . . . 9 $/\$
10	9	recnd 7794	. . . . . . . 8 $/\$
11		mulcl 7747	. . . . . . . 8 $/\$
12	7, 10, 11	sylancr 410	. . . . . . 7 $/\$
13	6, 12	addcld 7785	. . . . . 6 $/\$
14		recl 10625	. . . . . . . 8
15	14	adantl 275	. . . . . . 7 $/\$
16	15	recnd 7794	. . . . . 6 $/\$
17		imcl 10626	. . . . . . . . 9
18	17	adantl 275	. . . . . . . 8 $/\$
19	18	recnd 7794	. . . . . . 7 $/\$
20		mulcl 7747	. . . . . . 7 $/\$
21	7, 19, 20	sylancr 410	. . . . . 6 $/\$
22	13, 16, 21	adddid 7790	. . . . 5 $/\$
23	6, 12, 16	adddird 7791	. . . . . . 7 $/\$
24	6, 12, 21	adddird 7791	. . . . . . 7 $/\$
25	23, 24	oveq12d 5792	. . . . . 6 $/\$
26	5, 15	remulcld 7796	. . . . . . . 8 $/\$
27	26	recnd 7794	. . . . . . 7 $/\$
28	12, 21	mulcld 7786	. . . . . . 7 $/\$
29	12, 16	mulcld 7786	. . . . . . 7 $/\$
30	6, 21	mulcld 7786	. . . . . . 7 $/\$
31	27, 28, 29, 30	add42d 7932	. . . . . 6 $/\$
32	7	a1i 9	. . . . . . . . . . 11 $/\$
33	32, 10, 32, 19	mul4d 7917	. . . . . . . . . 10 $/\$
34		ixi 8345	. . . . . . . . . . . 12
35	34	oveq1i 5784	. . . . . . . . . . 11
36	9, 18	remulcld 7796	. . . . . . . . . . . . 13 $/\$
37	36	recnd 7794	. . . . . . . . . . . 12 $/\$
38	37	mulm1d 8172	. . . . . . . . . . 11 $/\$
39	35, 38	syl5eq 2184	. . . . . . . . . 10 $/\$
40	33, 39	eqtrd 2172	. . . . . . . . 9 $/\$
41	40	oveq2d 5790	. . . . . . . 8 $/\$
42	27, 37	negsubd 8079	. . . . . . . 8 $/\$
43	41, 42	eqtrd 2172	. . . . . . 7 $/\$
44	9, 15	remulcld 7796	. . . . . . . . . . 11 $/\$
45	44	recnd 7794	. . . . . . . . . 10 $/\$
46		mulcl 7747	. . . . . . . . . 10 $/\$
47	7, 45, 46	sylancr 410	. . . . . . . . 9 $/\$
48	5, 18	remulcld 7796	. . . . . . . . . . 11 $/\$
49	48	recnd 7794	. . . . . . . . . 10 $/\$
50		mulcl 7747	. . . . . . . . . 10 $/\$
51	7, 49, 50	sylancr 410	. . . . . . . . 9 $/\$
52	47, 51	addcomd 7913	. . . . . . . 8 $/\$
53	32, 10, 16	mulassd 7789	. . . . . . . . 9 $/\$
54	6, 32, 19	mul12d 7914	. . . . . . . . 9 $/\$
55	53, 54	oveq12d 5792	. . . . . . . 8 $/\$
56	32, 49, 45	adddid 7790	. . . . . . . 8 $/\$
57	52, 55, 56	3eqtr4d 2182	. . . . . . 7 $/\$
58	43, 57	oveq12d 5792	. . . . . 6 $/\$
59	25, 31, 58	3eqtr2d 2178	. . . . 5 $/\$
60	3, 22, 59	3eqtrd 2176	. . . 4 $/\$
61	60	fveq2d 5425	. . 3 $/\$
62	26, 36	resubcld 8143	. . . 4 $/\$
63	48, 44	readdcld 7795	. . . 4 $/\$
64		crre 10629	. . . 4 $/\$
65	62, 63, 64	syl2anc 408	. . 3 $/\$
66	61, 65	eqtrd 2172	. 2 $/\$
67	60	fveq2d 5425	. . 3 $/\$
68		crim 10630	. . . 4 $/\$
69	62, 63, 68	syl2anc 408	. . 3 $/\$
70	67, 69	eqtrd 2172	. 2 $/\$
71		mulcl 7747	. . . 4 $/\$
72		remim 10632	. . . 4
73	71, 72	syl 14	. . 3 $/\$
74		remim 10632	. . . . 5
75		remim 10632	. . . . 5
76	74, 75	oveqan12d 5793	. . . 4 $/\$
77	16, 21	subcld 8073	. . . . 5 $/\$
78	6, 12, 77	subdird 8177	. . . 4 $/\$
79	27, 30, 29, 28	subadd4d 8121	. . . . 5 $/\$
80	6, 16, 21	subdid 8176	. . . . . 6 $/\$
81	12, 16, 21	subdid 8176	. . . . . 6 $/\$
82	80, 81	oveq12d 5792	. . . . 5 $/\$
83	65, 61, 43	3eqtr4d 2182	. . . . . 6 $/\$
84	70	oveq2d 5790	. . . . . . 7 $/\$
85	54, 53	oveq12d 5792	. . . . . . 7 $/\$
86	56, 84, 85	3eqtr4d 2182	. . . . . 6 $/\$
87	83, 86	oveq12d 5792	. . . . 5 $/\$
88	79, 82, 87	3eqtr4d 2182	. . . 4 $/\$
89	76, 78, 88	3eqtrd 2176	. . 3 $/\$
90	73, 89	eqtr4d 2175	. 2 $/\$
91	66, 70, 90	3jca 1161	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 962

wceq 1331

wcel 1480

cfv 5123 (class class class)co 5774

cc 7618

cr 7619

c1 7621

ci 7622

caddc 7623

cmul 7625

cmin 7933

cneg 7934

ccj 10611

cre 10612

cim 10613

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738

This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-po 4218 df-iso 4219 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-2 8779 df-cj 10614 df-re 10615 df-im 10616

This theorem is referenced by: remul 10644 immul 10651 cjmul 10657

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