remullem - Intuitionistic Logic Explorer

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Theorem remullem 10813

Description: Lemma for remul 10814, immul 10821, and cjmul 10827. (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 10801	. . . . . 6
2		replim 10801	. . . . . 6
3	1, 2	oveqan12d 5861	. . . . 5 $/\$
4		recl 10795	. . . . . . . . 9
5	4	adantr 274	. . . . . . . 8 $/\$
6	5	recnd 7927	. . . . . . 7 $/\$
7		ax-icn 7848	. . . . . . . 8
8		imcl 10796	. . . . . . . . . 10
9	8	adantr 274	. . . . . . . . 9 $/\$
10	9	recnd 7927	. . . . . . . 8 $/\$
11		mulcl 7880	. . . . . . . 8 $/\$
12	7, 10, 11	sylancr 411	. . . . . . 7 $/\$
13	6, 12	addcld 7918	. . . . . 6 $/\$
14		recl 10795	. . . . . . . 8
15	14	adantl 275	. . . . . . 7 $/\$
16	15	recnd 7927	. . . . . 6 $/\$
17		imcl 10796	. . . . . . . . 9
18	17	adantl 275	. . . . . . . 8 $/\$
19	18	recnd 7927	. . . . . . 7 $/\$
20		mulcl 7880	. . . . . . 7 $/\$
21	7, 19, 20	sylancr 411	. . . . . 6 $/\$
22	13, 16, 21	adddid 7923	. . . . 5 $/\$
23	6, 12, 16	adddird 7924	. . . . . . 7 $/\$
24	6, 12, 21	adddird 7924	. . . . . . 7 $/\$
25	23, 24	oveq12d 5860	. . . . . 6 $/\$
26	5, 15	remulcld 7929	. . . . . . . 8 $/\$
27	26	recnd 7927	. . . . . . 7 $/\$
28	12, 21	mulcld 7919	. . . . . . 7 $/\$
29	12, 16	mulcld 7919	. . . . . . 7 $/\$
30	6, 21	mulcld 7919	. . . . . . 7 $/\$
31	27, 28, 29, 30	add42d 8068	. . . . . 6 $/\$
32	7	a1i 9	. . . . . . . . . . 11 $/\$
33	32, 10, 32, 19	mul4d 8053	. . . . . . . . . 10 $/\$
34		ixi 8481	. . . . . . . . . . . 12
35	34	oveq1i 5852	. . . . . . . . . . 11
36	9, 18	remulcld 7929	. . . . . . . . . . . . 13 $/\$
37	36	recnd 7927	. . . . . . . . . . . 12 $/\$
38	37	mulm1d 8308	. . . . . . . . . . 11 $/\$
39	35, 38	syl5eq 2211	. . . . . . . . . 10 $/\$
40	33, 39	eqtrd 2198	. . . . . . . . 9 $/\$
41	40	oveq2d 5858	. . . . . . . 8 $/\$
42	27, 37	negsubd 8215	. . . . . . . 8 $/\$
43	41, 42	eqtrd 2198	. . . . . . 7 $/\$
44	9, 15	remulcld 7929	. . . . . . . . . . 11 $/\$
45	44	recnd 7927	. . . . . . . . . 10 $/\$
46		mulcl 7880	. . . . . . . . . 10 $/\$
47	7, 45, 46	sylancr 411	. . . . . . . . 9 $/\$
48	5, 18	remulcld 7929	. . . . . . . . . . 11 $/\$
49	48	recnd 7927	. . . . . . . . . 10 $/\$
50		mulcl 7880	. . . . . . . . . 10 $/\$
51	7, 49, 50	sylancr 411	. . . . . . . . 9 $/\$
52	47, 51	addcomd 8049	. . . . . . . 8 $/\$
53	32, 10, 16	mulassd 7922	. . . . . . . . 9 $/\$
54	6, 32, 19	mul12d 8050	. . . . . . . . 9 $/\$
55	53, 54	oveq12d 5860	. . . . . . . 8 $/\$
56	32, 49, 45	adddid 7923	. . . . . . . 8 $/\$
57	52, 55, 56	3eqtr4d 2208	. . . . . . 7 $/\$
58	43, 57	oveq12d 5860	. . . . . 6 $/\$
59	25, 31, 58	3eqtr2d 2204	. . . . 5 $/\$
60	3, 22, 59	3eqtrd 2202	. . . 4 $/\$
61	60	fveq2d 5490	. . 3 $/\$
62	26, 36	resubcld 8279	. . . 4 $/\$
63	48, 44	readdcld 7928	. . . 4 $/\$
64		crre 10799	. . . 4 $/\$
65	62, 63, 64	syl2anc 409	. . 3 $/\$
66	61, 65	eqtrd 2198	. 2 $/\$
67	60	fveq2d 5490	. . 3 $/\$
68		crim 10800	. . . 4 $/\$
69	62, 63, 68	syl2anc 409	. . 3 $/\$
70	67, 69	eqtrd 2198	. 2 $/\$
71		mulcl 7880	. . . 4 $/\$
72		remim 10802	. . . 4
73	71, 72	syl 14	. . 3 $/\$
74		remim 10802	. . . . 5
75		remim 10802	. . . . 5
76	74, 75	oveqan12d 5861	. . . 4 $/\$
77	16, 21	subcld 8209	. . . . 5 $/\$
78	6, 12, 77	subdird 8313	. . . 4 $/\$
79	27, 30, 29, 28	subadd4d 8257	. . . . 5 $/\$
80	6, 16, 21	subdid 8312	. . . . . 6 $/\$
81	12, 16, 21	subdid 8312	. . . . . 6 $/\$
82	80, 81	oveq12d 5860	. . . . 5 $/\$
83	65, 61, 43	3eqtr4d 2208	. . . . . 6 $/\$
84	70	oveq2d 5858	. . . . . . 7 $/\$
85	54, 53	oveq12d 5860	. . . . . . 7 $/\$
86	56, 84, 85	3eqtr4d 2208	. . . . . 6 $/\$
87	83, 86	oveq12d 5860	. . . . 5 $/\$
88	79, 82, 87	3eqtr4d 2208	. . . 4 $/\$
89	76, 78, 88	3eqtrd 2202	. . 3 $/\$
90	73, 89	eqtr4d 2201	. 2 $/\$
91	66, 70, 90	3jca 1167	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 968

wceq 1343

wcel 2136

cfv 5188 (class class class)co 5842

cc 7751

cr 7752

c1 7754

ci 7755

caddc 7756

cmul 7758

cmin 8069

cneg 8070

ccj 10781

cre 10782

cim 10783

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 ax-pre-mulext 7871

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rmo 2452 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-mpt 4045 df-id 4271 df-po 4274 df-iso 4275 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-rn 4615 df-res 4616 df-ima 4617 df-iota 5153 df-fun 5190 df-fn 5191 df-f 5192 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938 df-le 7939 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 df-div 8569 df-2 8916 df-cj 10784 df-re 10785 df-im 10786

This theorem is referenced by: remul 10814 immul 10821 cjmul 10827

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