remullem - Intuitionistic Logic Explorer

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Theorem remullem 10835

Description: Lemma for remul 10836, immul 10843, and cjmul 10849. (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 10823	. . . . . 6
2		replim 10823	. . . . . 6
3	1, 2	oveqan12d 5872	. . . . 5 $/\$
4		recl 10817	. . . . . . . . 9
5	4	adantr 274	. . . . . . . 8 $/\$
6	5	recnd 7948	. . . . . . 7 $/\$
7		ax-icn 7869	. . . . . . . 8
8		imcl 10818	. . . . . . . . . 10
9	8	adantr 274	. . . . . . . . 9 $/\$
10	9	recnd 7948	. . . . . . . 8 $/\$
11		mulcl 7901	. . . . . . . 8 $/\$
12	7, 10, 11	sylancr 412	. . . . . . 7 $/\$
13	6, 12	addcld 7939	. . . . . 6 $/\$
14		recl 10817	. . . . . . . 8
15	14	adantl 275	. . . . . . 7 $/\$
16	15	recnd 7948	. . . . . 6 $/\$
17		imcl 10818	. . . . . . . . 9
18	17	adantl 275	. . . . . . . 8 $/\$
19	18	recnd 7948	. . . . . . 7 $/\$
20		mulcl 7901	. . . . . . 7 $/\$
21	7, 19, 20	sylancr 412	. . . . . 6 $/\$
22	13, 16, 21	adddid 7944	. . . . 5 $/\$
23	6, 12, 16	adddird 7945	. . . . . . 7 $/\$
24	6, 12, 21	adddird 7945	. . . . . . 7 $/\$
25	23, 24	oveq12d 5871	. . . . . 6 $/\$
26	5, 15	remulcld 7950	. . . . . . . 8 $/\$
27	26	recnd 7948	. . . . . . 7 $/\$
28	12, 21	mulcld 7940	. . . . . . 7 $/\$
29	12, 16	mulcld 7940	. . . . . . 7 $/\$
30	6, 21	mulcld 7940	. . . . . . 7 $/\$
31	27, 28, 29, 30	add42d 8089	. . . . . 6 $/\$
32	7	a1i 9	. . . . . . . . . . 11 $/\$
33	32, 10, 32, 19	mul4d 8074	. . . . . . . . . 10 $/\$
34		ixi 8502	. . . . . . . . . . . 12
35	34	oveq1i 5863	. . . . . . . . . . 11
36	9, 18	remulcld 7950	. . . . . . . . . . . . 13 $/\$
37	36	recnd 7948	. . . . . . . . . . . 12 $/\$
38	37	mulm1d 8329	. . . . . . . . . . 11 $/\$
39	35, 38	eqtrid 2215	. . . . . . . . . 10 $/\$
40	33, 39	eqtrd 2203	. . . . . . . . 9 $/\$
41	40	oveq2d 5869	. . . . . . . 8 $/\$
42	27, 37	negsubd 8236	. . . . . . . 8 $/\$
43	41, 42	eqtrd 2203	. . . . . . 7 $/\$
44	9, 15	remulcld 7950	. . . . . . . . . . 11 $/\$
45	44	recnd 7948	. . . . . . . . . 10 $/\$
46		mulcl 7901	. . . . . . . . . 10 $/\$
47	7, 45, 46	sylancr 412	. . . . . . . . 9 $/\$
48	5, 18	remulcld 7950	. . . . . . . . . . 11 $/\$
49	48	recnd 7948	. . . . . . . . . 10 $/\$
50		mulcl 7901	. . . . . . . . . 10 $/\$
51	7, 49, 50	sylancr 412	. . . . . . . . 9 $/\$
52	47, 51	addcomd 8070	. . . . . . . 8 $/\$
53	32, 10, 16	mulassd 7943	. . . . . . . . 9 $/\$
54	6, 32, 19	mul12d 8071	. . . . . . . . 9 $/\$
55	53, 54	oveq12d 5871	. . . . . . . 8 $/\$
56	32, 49, 45	adddid 7944	. . . . . . . 8 $/\$
57	52, 55, 56	3eqtr4d 2213	. . . . . . 7 $/\$
58	43, 57	oveq12d 5871	. . . . . 6 $/\$
59	25, 31, 58	3eqtr2d 2209	. . . . 5 $/\$
60	3, 22, 59	3eqtrd 2207	. . . 4 $/\$
61	60	fveq2d 5500	. . 3 $/\$
62	26, 36	resubcld 8300	. . . 4 $/\$
63	48, 44	readdcld 7949	. . . 4 $/\$
64		crre 10821	. . . 4 $/\$
65	62, 63, 64	syl2anc 409	. . 3 $/\$
66	61, 65	eqtrd 2203	. 2 $/\$
67	60	fveq2d 5500	. . 3 $/\$
68		crim 10822	. . . 4 $/\$
69	62, 63, 68	syl2anc 409	. . 3 $/\$
70	67, 69	eqtrd 2203	. 2 $/\$
71		mulcl 7901	. . . 4 $/\$
72		remim 10824	. . . 4
73	71, 72	syl 14	. . 3 $/\$
74		remim 10824	. . . . 5
75		remim 10824	. . . . 5
76	74, 75	oveqan12d 5872	. . . 4 $/\$
77	16, 21	subcld 8230	. . . . 5 $/\$
78	6, 12, 77	subdird 8334	. . . 4 $/\$
79	27, 30, 29, 28	subadd4d 8278	. . . . 5 $/\$
80	6, 16, 21	subdid 8333	. . . . . 6 $/\$
81	12, 16, 21	subdid 8333	. . . . . 6 $/\$
82	80, 81	oveq12d 5871	. . . . 5 $/\$
83	65, 61, 43	3eqtr4d 2213	. . . . . 6 $/\$
84	70	oveq2d 5869	. . . . . . 7 $/\$
85	54, 53	oveq12d 5871	. . . . . . 7 $/\$
86	56, 84, 85	3eqtr4d 2213	. . . . . 6 $/\$
87	83, 86	oveq12d 5871	. . . . 5 $/\$
88	79, 82, 87	3eqtr4d 2213	. . . 4 $/\$
89	76, 78, 88	3eqtrd 2207	. . 3 $/\$
90	73, 89	eqtr4d 2206	. 2 $/\$
91	66, 70, 90	3jca 1172	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 973

wceq 1348

wcel 2141

cfv 5198 (class class class)co 5853

cc 7772

cr 7773

c1 7775

ci 7776

caddc 7777

cmul 7779

cmin 8090

cneg 8091

ccj 10803

cre 10804

cim 10805

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-po 4281 df-iso 4282 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-2 8937 df-cj 10806 df-re 10807 df-im 10808

This theorem is referenced by: remul 10836 immul 10843 cjmul 10849

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