remullem - Intuitionistic Logic Explorer

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Theorem remullem 10799

Description: Lemma for remul 10800, immul 10807, and cjmul 10813. (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 10787	. . . . . 6
2		replim 10787	. . . . . 6
3	1, 2	oveqan12d 5855	. . . . 5 $/\$
4		recl 10781	. . . . . . . . 9
5	4	adantr 274	. . . . . . . 8 $/\$
6	5	recnd 7918	. . . . . . 7 $/\$
7		ax-icn 7839	. . . . . . . 8
8		imcl 10782	. . . . . . . . . 10
9	8	adantr 274	. . . . . . . . 9 $/\$
10	9	recnd 7918	. . . . . . . 8 $/\$
11		mulcl 7871	. . . . . . . 8 $/\$
12	7, 10, 11	sylancr 411	. . . . . . 7 $/\$
13	6, 12	addcld 7909	. . . . . 6 $/\$
14		recl 10781	. . . . . . . 8
15	14	adantl 275	. . . . . . 7 $/\$
16	15	recnd 7918	. . . . . 6 $/\$
17		imcl 10782	. . . . . . . . 9
18	17	adantl 275	. . . . . . . 8 $/\$
19	18	recnd 7918	. . . . . . 7 $/\$
20		mulcl 7871	. . . . . . 7 $/\$
21	7, 19, 20	sylancr 411	. . . . . 6 $/\$
22	13, 16, 21	adddid 7914	. . . . 5 $/\$
23	6, 12, 16	adddird 7915	. . . . . . 7 $/\$
24	6, 12, 21	adddird 7915	. . . . . . 7 $/\$
25	23, 24	oveq12d 5854	. . . . . 6 $/\$
26	5, 15	remulcld 7920	. . . . . . . 8 $/\$
27	26	recnd 7918	. . . . . . 7 $/\$
28	12, 21	mulcld 7910	. . . . . . 7 $/\$
29	12, 16	mulcld 7910	. . . . . . 7 $/\$
30	6, 21	mulcld 7910	. . . . . . 7 $/\$
31	27, 28, 29, 30	add42d 8059	. . . . . 6 $/\$
32	7	a1i 9	. . . . . . . . . . 11 $/\$
33	32, 10, 32, 19	mul4d 8044	. . . . . . . . . 10 $/\$
34		ixi 8472	. . . . . . . . . . . 12
35	34	oveq1i 5846	. . . . . . . . . . 11
36	9, 18	remulcld 7920	. . . . . . . . . . . . 13 $/\$
37	36	recnd 7918	. . . . . . . . . . . 12 $/\$
38	37	mulm1d 8299	. . . . . . . . . . 11 $/\$
39	35, 38	syl5eq 2209	. . . . . . . . . 10 $/\$
40	33, 39	eqtrd 2197	. . . . . . . . 9 $/\$
41	40	oveq2d 5852	. . . . . . . 8 $/\$
42	27, 37	negsubd 8206	. . . . . . . 8 $/\$
43	41, 42	eqtrd 2197	. . . . . . 7 $/\$
44	9, 15	remulcld 7920	. . . . . . . . . . 11 $/\$
45	44	recnd 7918	. . . . . . . . . 10 $/\$
46		mulcl 7871	. . . . . . . . . 10 $/\$
47	7, 45, 46	sylancr 411	. . . . . . . . 9 $/\$
48	5, 18	remulcld 7920	. . . . . . . . . . 11 $/\$
49	48	recnd 7918	. . . . . . . . . 10 $/\$
50		mulcl 7871	. . . . . . . . . 10 $/\$
51	7, 49, 50	sylancr 411	. . . . . . . . 9 $/\$
52	47, 51	addcomd 8040	. . . . . . . 8 $/\$
53	32, 10, 16	mulassd 7913	. . . . . . . . 9 $/\$
54	6, 32, 19	mul12d 8041	. . . . . . . . 9 $/\$
55	53, 54	oveq12d 5854	. . . . . . . 8 $/\$
56	32, 49, 45	adddid 7914	. . . . . . . 8 $/\$
57	52, 55, 56	3eqtr4d 2207	. . . . . . 7 $/\$
58	43, 57	oveq12d 5854	. . . . . 6 $/\$
59	25, 31, 58	3eqtr2d 2203	. . . . 5 $/\$
60	3, 22, 59	3eqtrd 2201	. . . 4 $/\$
61	60	fveq2d 5484	. . 3 $/\$
62	26, 36	resubcld 8270	. . . 4 $/\$
63	48, 44	readdcld 7919	. . . 4 $/\$
64		crre 10785	. . . 4 $/\$
65	62, 63, 64	syl2anc 409	. . 3 $/\$
66	61, 65	eqtrd 2197	. 2 $/\$
67	60	fveq2d 5484	. . 3 $/\$
68		crim 10786	. . . 4 $/\$
69	62, 63, 68	syl2anc 409	. . 3 $/\$
70	67, 69	eqtrd 2197	. 2 $/\$
71		mulcl 7871	. . . 4 $/\$
72		remim 10788	. . . 4
73	71, 72	syl 14	. . 3 $/\$
74		remim 10788	. . . . 5
75		remim 10788	. . . . 5
76	74, 75	oveqan12d 5855	. . . 4 $/\$
77	16, 21	subcld 8200	. . . . 5 $/\$
78	6, 12, 77	subdird 8304	. . . 4 $/\$
79	27, 30, 29, 28	subadd4d 8248	. . . . 5 $/\$
80	6, 16, 21	subdid 8303	. . . . . 6 $/\$
81	12, 16, 21	subdid 8303	. . . . . 6 $/\$
82	80, 81	oveq12d 5854	. . . . 5 $/\$
83	65, 61, 43	3eqtr4d 2207	. . . . . 6 $/\$
84	70	oveq2d 5852	. . . . . . 7 $/\$
85	54, 53	oveq12d 5854	. . . . . . 7 $/\$
86	56, 84, 85	3eqtr4d 2207	. . . . . 6 $/\$
87	83, 86	oveq12d 5854	. . . . 5 $/\$
88	79, 82, 87	3eqtr4d 2207	. . . 4 $/\$
89	76, 78, 88	3eqtrd 2201	. . 3 $/\$
90	73, 89	eqtr4d 2200	. 2 $/\$
91	66, 70, 90	3jca 1166	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 103 $/\$ w3a 967

wceq 1342

wcel 2135

cfv 5182 (class class class)co 5836

cc 7742

cr 7743

c1 7745

ci 7746

caddc 7747

cmul 7749

cmin 8060

cneg 8061

ccj 10767

cre 10768

cim 10769

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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulrcl 7843 ax-addcom 7844 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-i2m1 7849 ax-0lt1 7850 ax-1rid 7851 ax-0id 7852 ax-rnegex 7853 ax-precex 7854 ax-cnre 7855 ax-pre-ltirr 7856 ax-pre-ltwlin 7857 ax-pre-lttrn 7858 ax-pre-apti 7859 ax-pre-ltadd 7860 ax-pre-mulgt0 7861 ax-pre-mulext 7862

This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-po 4268 df-iso 4269 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fv 5190 df-riota 5792 df-ov 5839 df-oprab 5840 df-mpo 5841 df-pnf 7926 df-mnf 7927 df-xr 7928 df-ltxr 7929 df-le 7930 df-sub 8062 df-neg 8063 df-reap 8464 df-ap 8471 df-div 8560 df-2 8907 df-cj 10770 df-re 10771 df-im 10772

This theorem is referenced by: remul 10800 immul 10807 cjmul 10813

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