remullem - Intuitionistic Logic Explorer

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Theorem remullem 10864

Description: Lemma for remul 10865, immul 10872, and cjmul 10878. (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 10852	. . . . . 6
2		replim 10852	. . . . . 6
3	1, 2	oveqan12d 5888	. . . . 5 $/\$
4		recl 10846	. . . . . . . . 9
5	4	adantr 276	. . . . . . . 8 $/\$
6	5	recnd 7976	. . . . . . 7 $/\$
7		ax-icn 7897	. . . . . . . 8
8		imcl 10847	. . . . . . . . . 10
9	8	adantr 276	. . . . . . . . 9 $/\$
10	9	recnd 7976	. . . . . . . 8 $/\$
11		mulcl 7929	. . . . . . . 8 $/\$
12	7, 10, 11	sylancr 414	. . . . . . 7 $/\$
13	6, 12	addcld 7967	. . . . . 6 $/\$
14		recl 10846	. . . . . . . 8
15	14	adantl 277	. . . . . . 7 $/\$
16	15	recnd 7976	. . . . . 6 $/\$
17		imcl 10847	. . . . . . . . 9
18	17	adantl 277	. . . . . . . 8 $/\$
19	18	recnd 7976	. . . . . . 7 $/\$
20		mulcl 7929	. . . . . . 7 $/\$
21	7, 19, 20	sylancr 414	. . . . . 6 $/\$
22	13, 16, 21	adddid 7972	. . . . 5 $/\$
23	6, 12, 16	adddird 7973	. . . . . . 7 $/\$
24	6, 12, 21	adddird 7973	. . . . . . 7 $/\$
25	23, 24	oveq12d 5887	. . . . . 6 $/\$
26	5, 15	remulcld 7978	. . . . . . . 8 $/\$
27	26	recnd 7976	. . . . . . 7 $/\$
28	12, 21	mulcld 7968	. . . . . . 7 $/\$
29	12, 16	mulcld 7968	. . . . . . 7 $/\$
30	6, 21	mulcld 7968	. . . . . . 7 $/\$
31	27, 28, 29, 30	add42d 8117	. . . . . 6 $/\$
32	7	a1i 9	. . . . . . . . . . 11 $/\$
33	32, 10, 32, 19	mul4d 8102	. . . . . . . . . 10 $/\$
34		ixi 8530	. . . . . . . . . . . 12
35	34	oveq1i 5879	. . . . . . . . . . 11
36	9, 18	remulcld 7978	. . . . . . . . . . . . 13 $/\$
37	36	recnd 7976	. . . . . . . . . . . 12 $/\$
38	37	mulm1d 8357	. . . . . . . . . . 11 $/\$
39	35, 38	eqtrid 2222	. . . . . . . . . 10 $/\$
40	33, 39	eqtrd 2210	. . . . . . . . 9 $/\$
41	40	oveq2d 5885	. . . . . . . 8 $/\$
42	27, 37	negsubd 8264	. . . . . . . 8 $/\$
43	41, 42	eqtrd 2210	. . . . . . 7 $/\$
44	9, 15	remulcld 7978	. . . . . . . . . . 11 $/\$
45	44	recnd 7976	. . . . . . . . . 10 $/\$
46		mulcl 7929	. . . . . . . . . 10 $/\$
47	7, 45, 46	sylancr 414	. . . . . . . . 9 $/\$
48	5, 18	remulcld 7978	. . . . . . . . . . 11 $/\$
49	48	recnd 7976	. . . . . . . . . 10 $/\$
50		mulcl 7929	. . . . . . . . . 10 $/\$
51	7, 49, 50	sylancr 414	. . . . . . . . 9 $/\$
52	47, 51	addcomd 8098	. . . . . . . 8 $/\$
53	32, 10, 16	mulassd 7971	. . . . . . . . 9 $/\$
54	6, 32, 19	mul12d 8099	. . . . . . . . 9 $/\$
55	53, 54	oveq12d 5887	. . . . . . . 8 $/\$
56	32, 49, 45	adddid 7972	. . . . . . . 8 $/\$
57	52, 55, 56	3eqtr4d 2220	. . . . . . 7 $/\$
58	43, 57	oveq12d 5887	. . . . . 6 $/\$
59	25, 31, 58	3eqtr2d 2216	. . . . 5 $/\$
60	3, 22, 59	3eqtrd 2214	. . . 4 $/\$
61	60	fveq2d 5515	. . 3 $/\$
62	26, 36	resubcld 8328	. . . 4 $/\$
63	48, 44	readdcld 7977	. . . 4 $/\$
64		crre 10850	. . . 4 $/\$
65	62, 63, 64	syl2anc 411	. . 3 $/\$
66	61, 65	eqtrd 2210	. 2 $/\$
67	60	fveq2d 5515	. . 3 $/\$
68		crim 10851	. . . 4 $/\$
69	62, 63, 68	syl2anc 411	. . 3 $/\$
70	67, 69	eqtrd 2210	. 2 $/\$
71		mulcl 7929	. . . 4 $/\$
72		remim 10853	. . . 4
73	71, 72	syl 14	. . 3 $/\$
74		remim 10853	. . . . 5
75		remim 10853	. . . . 5
76	74, 75	oveqan12d 5888	. . . 4 $/\$
77	16, 21	subcld 8258	. . . . 5 $/\$
78	6, 12, 77	subdird 8362	. . . 4 $/\$
79	27, 30, 29, 28	subadd4d 8306	. . . . 5 $/\$
80	6, 16, 21	subdid 8361	. . . . . 6 $/\$
81	12, 16, 21	subdid 8361	. . . . . 6 $/\$
82	80, 81	oveq12d 5887	. . . . 5 $/\$
83	65, 61, 43	3eqtr4d 2220	. . . . . 6 $/\$
84	70	oveq2d 5885	. . . . . . 7 $/\$
85	54, 53	oveq12d 5887	. . . . . . 7 $/\$
86	56, 84, 85	3eqtr4d 2220	. . . . . 6 $/\$
87	83, 86	oveq12d 5887	. . . . 5 $/\$
88	79, 82, 87	3eqtr4d 2220	. . . 4 $/\$
89	76, 78, 88	3eqtrd 2214	. . 3 $/\$
90	73, 89	eqtr4d 2213	. 2 $/\$
91	66, 70, 90	3jca 1177	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 978

wceq 1353

wcel 2148

cfv 5212 (class class class)co 5869

cc 7800

cr 7801

c1 7803

ci 7804

caddc 7805

cmul 7807

cmin 8118

cneg 8119

ccj 10832

cre 10833

cim 10834

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920

This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-2 8967 df-cj 10835 df-re 10836 df-im 10837

This theorem is referenced by: remul 10865 immul 10872 cjmul 10878

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