remullem - Intuitionistic Logic Explorer

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Theorem remullem 10882

Description: Lemma for remul 10883, immul 10890, and cjmul 10896. (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 10870	. . . . . 6
2		replim 10870	. . . . . 6
3	1, 2	oveqan12d 5896	. . . . 5 $/\$
4		recl 10864	. . . . . . . . 9
5	4	adantr 276	. . . . . . . 8 $/\$
6	5	recnd 7988	. . . . . . 7 $/\$
7		ax-icn 7908	. . . . . . . 8
8		imcl 10865	. . . . . . . . . 10
9	8	adantr 276	. . . . . . . . 9 $/\$
10	9	recnd 7988	. . . . . . . 8 $/\$
11		mulcl 7940	. . . . . . . 8 $/\$
12	7, 10, 11	sylancr 414	. . . . . . 7 $/\$
13	6, 12	addcld 7979	. . . . . 6 $/\$
14		recl 10864	. . . . . . . 8
15	14	adantl 277	. . . . . . 7 $/\$
16	15	recnd 7988	. . . . . 6 $/\$
17		imcl 10865	. . . . . . . . 9
18	17	adantl 277	. . . . . . . 8 $/\$
19	18	recnd 7988	. . . . . . 7 $/\$
20		mulcl 7940	. . . . . . 7 $/\$
21	7, 19, 20	sylancr 414	. . . . . 6 $/\$
22	13, 16, 21	adddid 7984	. . . . 5 $/\$
23	6, 12, 16	adddird 7985	. . . . . . 7 $/\$
24	6, 12, 21	adddird 7985	. . . . . . 7 $/\$
25	23, 24	oveq12d 5895	. . . . . 6 $/\$
26	5, 15	remulcld 7990	. . . . . . . 8 $/\$
27	26	recnd 7988	. . . . . . 7 $/\$
28	12, 21	mulcld 7980	. . . . . . 7 $/\$
29	12, 16	mulcld 7980	. . . . . . 7 $/\$
30	6, 21	mulcld 7980	. . . . . . 7 $/\$
31	27, 28, 29, 30	add42d 8129	. . . . . 6 $/\$
32	7	a1i 9	. . . . . . . . . . 11 $/\$
33	32, 10, 32, 19	mul4d 8114	. . . . . . . . . 10 $/\$
34		ixi 8542	. . . . . . . . . . . 12
35	34	oveq1i 5887	. . . . . . . . . . 11
36	9, 18	remulcld 7990	. . . . . . . . . . . . 13 $/\$
37	36	recnd 7988	. . . . . . . . . . . 12 $/\$
38	37	mulm1d 8369	. . . . . . . . . . 11 $/\$
39	35, 38	eqtrid 2222	. . . . . . . . . 10 $/\$
40	33, 39	eqtrd 2210	. . . . . . . . 9 $/\$
41	40	oveq2d 5893	. . . . . . . 8 $/\$
42	27, 37	negsubd 8276	. . . . . . . 8 $/\$
43	41, 42	eqtrd 2210	. . . . . . 7 $/\$
44	9, 15	remulcld 7990	. . . . . . . . . . 11 $/\$
45	44	recnd 7988	. . . . . . . . . 10 $/\$
46		mulcl 7940	. . . . . . . . . 10 $/\$
47	7, 45, 46	sylancr 414	. . . . . . . . 9 $/\$
48	5, 18	remulcld 7990	. . . . . . . . . . 11 $/\$
49	48	recnd 7988	. . . . . . . . . 10 $/\$
50		mulcl 7940	. . . . . . . . . 10 $/\$
51	7, 49, 50	sylancr 414	. . . . . . . . 9 $/\$
52	47, 51	addcomd 8110	. . . . . . . 8 $/\$
53	32, 10, 16	mulassd 7983	. . . . . . . . 9 $/\$
54	6, 32, 19	mul12d 8111	. . . . . . . . 9 $/\$
55	53, 54	oveq12d 5895	. . . . . . . 8 $/\$
56	32, 49, 45	adddid 7984	. . . . . . . 8 $/\$
57	52, 55, 56	3eqtr4d 2220	. . . . . . 7 $/\$
58	43, 57	oveq12d 5895	. . . . . 6 $/\$
59	25, 31, 58	3eqtr2d 2216	. . . . 5 $/\$
60	3, 22, 59	3eqtrd 2214	. . . 4 $/\$
61	60	fveq2d 5521	. . 3 $/\$
62	26, 36	resubcld 8340	. . . 4 $/\$
63	48, 44	readdcld 7989	. . . 4 $/\$
64		crre 10868	. . . 4 $/\$
65	62, 63, 64	syl2anc 411	. . 3 $/\$
66	61, 65	eqtrd 2210	. 2 $/\$
67	60	fveq2d 5521	. . 3 $/\$
68		crim 10869	. . . 4 $/\$
69	62, 63, 68	syl2anc 411	. . 3 $/\$
70	67, 69	eqtrd 2210	. 2 $/\$
71		mulcl 7940	. . . 4 $/\$
72		remim 10871	. . . 4
73	71, 72	syl 14	. . 3 $/\$
74		remim 10871	. . . . 5
75		remim 10871	. . . . 5
76	74, 75	oveqan12d 5896	. . . 4 $/\$
77	16, 21	subcld 8270	. . . . 5 $/\$
78	6, 12, 77	subdird 8374	. . . 4 $/\$
79	27, 30, 29, 28	subadd4d 8318	. . . . 5 $/\$
80	6, 16, 21	subdid 8373	. . . . . 6 $/\$
81	12, 16, 21	subdid 8373	. . . . . 6 $/\$
82	80, 81	oveq12d 5895	. . . . 5 $/\$
83	65, 61, 43	3eqtr4d 2220	. . . . . 6 $/\$
84	70	oveq2d 5893	. . . . . . 7 $/\$
85	54, 53	oveq12d 5895	. . . . . . 7 $/\$
86	56, 84, 85	3eqtr4d 2220	. . . . . 6 $/\$
87	83, 86	oveq12d 5895	. . . . 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 5218 (class class class)co 5877

cc 7811

cr 7812

c1 7814

ci 7815

caddc 7816

cmul 7818

cmin 8130

cneg 8131

ccj 10850

cre 10851

cim 10852

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 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7904 ax-resscn 7905 ax-1cn 7906 ax-1re 7907 ax-icn 7908 ax-addcl 7909 ax-addrcl 7910 ax-mulcl 7911 ax-mulrcl 7912 ax-addcom 7913 ax-mulcom 7914 ax-addass 7915 ax-mulass 7916 ax-distr 7917 ax-i2m1 7918 ax-0lt1 7919 ax-1rid 7920 ax-0id 7921 ax-rnegex 7922 ax-precex 7923 ax-cnre 7924 ax-pre-ltirr 7925 ax-pre-ltwlin 7926 ax-pre-lttrn 7927 ax-pre-apti 7928 ax-pre-ltadd 7929 ax-pre-mulgt0 7930 ax-pre-mulext 7931

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 2741 df-sbc 2965 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-po 4298 df-iso 4299 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 df-riota 5833 df-ov 5880 df-oprab 5881 df-mpo 5882 df-pnf 7996 df-mnf 7997 df-xr 7998 df-ltxr 7999 df-le 8000 df-sub 8132 df-neg 8133 df-reap 8534 df-ap 8541 df-div 8632 df-2 8980 df-cj 10853 df-re 10854 df-im 10855

This theorem is referenced by: remul 10883 immul 10890 cjmul 10896

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