remullem - Intuitionistic Logic Explorer

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Theorem remullem 9758

Description: Lemma for remul 9759, immul 9766, and cjmul 9772. (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 9746	. . . . . 6
2		replim 9746	. . . . . 6
3	1, 2	oveqan12d 5551	. . . . 5 $/\$
4		recl 9740	. . . . . . . . 9
5	4	adantr 270	. . . . . . . 8 $/\$
6	5	recnd 7147	. . . . . . 7 $/\$
7		ax-icn 7071	. . . . . . . 8
8		imcl 9741	. . . . . . . . . 10
9	8	adantr 270	. . . . . . . . 9 $/\$
10	9	recnd 7147	. . . . . . . 8 $/\$
11		mulcl 7100	. . . . . . . 8 $/\$
12	7, 10, 11	sylancr 405	. . . . . . 7 $/\$
13	6, 12	addcld 7138	. . . . . 6 $/\$
14		recl 9740	. . . . . . . 8
15	14	adantl 271	. . . . . . 7 $/\$
16	15	recnd 7147	. . . . . 6 $/\$
17		imcl 9741	. . . . . . . . 9
18	17	adantl 271	. . . . . . . 8 $/\$
19	18	recnd 7147	. . . . . . 7 $/\$
20		mulcl 7100	. . . . . . 7 $/\$
21	7, 19, 20	sylancr 405	. . . . . 6 $/\$
22	13, 16, 21	adddid 7143	. . . . 5 $/\$
23	6, 12, 16	adddird 7144	. . . . . . 7 $/\$
24	6, 12, 21	adddird 7144	. . . . . . 7 $/\$
25	23, 24	oveq12d 5550	. . . . . 6 $/\$
26	5, 15	remulcld 7149	. . . . . . . 8 $/\$
27	26	recnd 7147	. . . . . . 7 $/\$
28	12, 21	mulcld 7139	. . . . . . 7 $/\$
29	12, 16	mulcld 7139	. . . . . . 7 $/\$
30	6, 21	mulcld 7139	. . . . . . 7 $/\$
31	27, 28, 29, 30	add42d 7278	. . . . . 6 $/\$
32	7	a1i 9	. . . . . . . . . . 11 $/\$
33	32, 10, 32, 19	mul4d 7263	. . . . . . . . . 10 $/\$
34		ixi 7683	. . . . . . . . . . . 12
35	34	oveq1i 5542	. . . . . . . . . . 11
36	9, 18	remulcld 7149	. . . . . . . . . . . . 13 $/\$
37	36	recnd 7147	. . . . . . . . . . . 12 $/\$
38	37	mulm1d 7514	. . . . . . . . . . 11 $/\$
39	35, 38	syl5eq 2125	. . . . . . . . . 10 $/\$
40	33, 39	eqtrd 2113	. . . . . . . . 9 $/\$
41	40	oveq2d 5548	. . . . . . . 8 $/\$
42	27, 37	negsubd 7425	. . . . . . . 8 $/\$
43	41, 42	eqtrd 2113	. . . . . . 7 $/\$
44	9, 15	remulcld 7149	. . . . . . . . . . 11 $/\$
45	44	recnd 7147	. . . . . . . . . 10 $/\$
46		mulcl 7100	. . . . . . . . . 10 $/\$
47	7, 45, 46	sylancr 405	. . . . . . . . 9 $/\$
48	5, 18	remulcld 7149	. . . . . . . . . . 11 $/\$
49	48	recnd 7147	. . . . . . . . . 10 $/\$
50		mulcl 7100	. . . . . . . . . 10 $/\$
51	7, 49, 50	sylancr 405	. . . . . . . . 9 $/\$
52	47, 51	addcomd 7259	. . . . . . . 8 $/\$
53	32, 10, 16	mulassd 7142	. . . . . . . . 9 $/\$
54	6, 32, 19	mul12d 7260	. . . . . . . . 9 $/\$
55	53, 54	oveq12d 5550	. . . . . . . 8 $/\$
56	32, 49, 45	adddid 7143	. . . . . . . 8 $/\$
57	52, 55, 56	3eqtr4d 2123	. . . . . . 7 $/\$
58	43, 57	oveq12d 5550	. . . . . 6 $/\$
59	25, 31, 58	3eqtr2d 2119	. . . . 5 $/\$
60	3, 22, 59	3eqtrd 2117	. . . 4 $/\$
61	60	fveq2d 5202	. . 3 $/\$
62	26, 36	resubcld 7485	. . . 4 $/\$
63	48, 44	readdcld 7148	. . . 4 $/\$
64		crre 9744	. . . 4 $/\$
65	62, 63, 64	syl2anc 403	. . 3 $/\$
66	61, 65	eqtrd 2113	. 2 $/\$
67	60	fveq2d 5202	. . 3 $/\$
68		crim 9745	. . . 4 $/\$
69	62, 63, 68	syl2anc 403	. . 3 $/\$
70	67, 69	eqtrd 2113	. 2 $/\$
71		mulcl 7100	. . . 4 $/\$
72		remim 9747	. . . 4
73	71, 72	syl 14	. . 3 $/\$
74		remim 9747	. . . . 5
75		remim 9747	. . . . 5
76	74, 75	oveqan12d 5551	. . . 4 $/\$
77	16, 21	subcld 7419	. . . . 5 $/\$
78	6, 12, 77	subdird 7519	. . . 4 $/\$
79	27, 30, 29, 28	subadd4d 7467	. . . . 5 $/\$
80	6, 16, 21	subdid 7518	. . . . . 6 $/\$
81	12, 16, 21	subdid 7518	. . . . . 6 $/\$
82	80, 81	oveq12d 5550	. . . . 5 $/\$
83	65, 61, 43	3eqtr4d 2123	. . . . . 6 $/\$
84	70	oveq2d 5548	. . . . . . 7 $/\$
85	54, 53	oveq12d 5550	. . . . . . 7 $/\$
86	56, 84, 85	3eqtr4d 2123	. . . . . 6 $/\$
87	83, 86	oveq12d 5550	. . . . 5 $/\$
88	79, 82, 87	3eqtr4d 2123	. . . 4 $/\$
89	76, 78, 88	3eqtrd 2117	. . 3 $/\$
90	73, 89	eqtr4d 2116	. 2 $/\$
91	66, 70, 90	3jca 1118	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 102 $/\$ w3a 919

wceq 1284

wcel 1433

cfv 4922 (class class class)co 5532

cc 6979

cr 6980

c1 6982

ci 6983

caddc 6984

cmul 6986

cmin 7279

cneg 7280

ccj 9726

cre 9727

cim 9728

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 ax-pre-mulext 7094

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 df-div 7761 df-2 8098 df-cj 9729 df-re 9730 df-im 9731

This theorem is referenced by: remul 9759 immul 9766 cjmul 9772

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