remullem - Intuitionistic Logic Explorer

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Theorem remullem 10822

Description: Lemma for remul 10823, immul 10830, and cjmul 10836. (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 10810	. . . . . 6
2		replim 10810	. . . . . 6
3	1, 2	oveqan12d 5869	. . . . 5 $/\$
4		recl 10804	. . . . . . . . 9
5	4	adantr 274	. . . . . . . 8 $/\$
6	5	recnd 7935	. . . . . . 7 $/\$
7		ax-icn 7856	. . . . . . . 8
8		imcl 10805	. . . . . . . . . 10
9	8	adantr 274	. . . . . . . . 9 $/\$
10	9	recnd 7935	. . . . . . . 8 $/\$
11		mulcl 7888	. . . . . . . 8 $/\$
12	7, 10, 11	sylancr 412	. . . . . . 7 $/\$
13	6, 12	addcld 7926	. . . . . 6 $/\$
14		recl 10804	. . . . . . . 8
15	14	adantl 275	. . . . . . 7 $/\$
16	15	recnd 7935	. . . . . 6 $/\$
17		imcl 10805	. . . . . . . . 9
18	17	adantl 275	. . . . . . . 8 $/\$
19	18	recnd 7935	. . . . . . 7 $/\$
20		mulcl 7888	. . . . . . 7 $/\$
21	7, 19, 20	sylancr 412	. . . . . 6 $/\$
22	13, 16, 21	adddid 7931	. . . . 5 $/\$
23	6, 12, 16	adddird 7932	. . . . . . 7 $/\$
24	6, 12, 21	adddird 7932	. . . . . . 7 $/\$
25	23, 24	oveq12d 5868	. . . . . 6 $/\$
26	5, 15	remulcld 7937	. . . . . . . 8 $/\$
27	26	recnd 7935	. . . . . . 7 $/\$
28	12, 21	mulcld 7927	. . . . . . 7 $/\$
29	12, 16	mulcld 7927	. . . . . . 7 $/\$
30	6, 21	mulcld 7927	. . . . . . 7 $/\$
31	27, 28, 29, 30	add42d 8076	. . . . . 6 $/\$
32	7	a1i 9	. . . . . . . . . . 11 $/\$
33	32, 10, 32, 19	mul4d 8061	. . . . . . . . . 10 $/\$
34		ixi 8489	. . . . . . . . . . . 12
35	34	oveq1i 5860	. . . . . . . . . . 11
36	9, 18	remulcld 7937	. . . . . . . . . . . . 13 $/\$
37	36	recnd 7935	. . . . . . . . . . . 12 $/\$
38	37	mulm1d 8316	. . . . . . . . . . 11 $/\$
39	35, 38	eqtrid 2215	. . . . . . . . . 10 $/\$
40	33, 39	eqtrd 2203	. . . . . . . . 9 $/\$
41	40	oveq2d 5866	. . . . . . . 8 $/\$
42	27, 37	negsubd 8223	. . . . . . . 8 $/\$
43	41, 42	eqtrd 2203	. . . . . . 7 $/\$
44	9, 15	remulcld 7937	. . . . . . . . . . 11 $/\$
45	44	recnd 7935	. . . . . . . . . 10 $/\$
46		mulcl 7888	. . . . . . . . . 10 $/\$
47	7, 45, 46	sylancr 412	. . . . . . . . 9 $/\$
48	5, 18	remulcld 7937	. . . . . . . . . . 11 $/\$
49	48	recnd 7935	. . . . . . . . . 10 $/\$
50		mulcl 7888	. . . . . . . . . 10 $/\$
51	7, 49, 50	sylancr 412	. . . . . . . . 9 $/\$
52	47, 51	addcomd 8057	. . . . . . . 8 $/\$
53	32, 10, 16	mulassd 7930	. . . . . . . . 9 $/\$
54	6, 32, 19	mul12d 8058	. . . . . . . . 9 $/\$
55	53, 54	oveq12d 5868	. . . . . . . 8 $/\$
56	32, 49, 45	adddid 7931	. . . . . . . 8 $/\$
57	52, 55, 56	3eqtr4d 2213	. . . . . . 7 $/\$
58	43, 57	oveq12d 5868	. . . . . 6 $/\$
59	25, 31, 58	3eqtr2d 2209	. . . . 5 $/\$
60	3, 22, 59	3eqtrd 2207	. . . 4 $/\$
61	60	fveq2d 5498	. . 3 $/\$
62	26, 36	resubcld 8287	. . . 4 $/\$
63	48, 44	readdcld 7936	. . . 4 $/\$
64		crre 10808	. . . 4 $/\$
65	62, 63, 64	syl2anc 409	. . 3 $/\$
66	61, 65	eqtrd 2203	. 2 $/\$
67	60	fveq2d 5498	. . 3 $/\$
68		crim 10809	. . . 4 $/\$
69	62, 63, 68	syl2anc 409	. . 3 $/\$
70	67, 69	eqtrd 2203	. 2 $/\$
71		mulcl 7888	. . . 4 $/\$
72		remim 10811	. . . 4
73	71, 72	syl 14	. . 3 $/\$
74		remim 10811	. . . . 5
75		remim 10811	. . . . 5
76	74, 75	oveqan12d 5869	. . . 4 $/\$
77	16, 21	subcld 8217	. . . . 5 $/\$
78	6, 12, 77	subdird 8321	. . . 4 $/\$
79	27, 30, 29, 28	subadd4d 8265	. . . . 5 $/\$
80	6, 16, 21	subdid 8320	. . . . . 6 $/\$
81	12, 16, 21	subdid 8320	. . . . . 6 $/\$
82	80, 81	oveq12d 5868	. . . . 5 $/\$
83	65, 61, 43	3eqtr4d 2213	. . . . . 6 $/\$
84	70	oveq2d 5866	. . . . . . 7 $/\$
85	54, 53	oveq12d 5868	. . . . . . 7 $/\$
86	56, 84, 85	3eqtr4d 2213	. . . . . 6 $/\$
87	83, 86	oveq12d 5868	. . . . 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 5196 (class class class)co 5850

cc 7759

cr 7760

c1 7762

ci 7763

caddc 7764

cmul 7766

cmin 8077

cneg 8078

ccj 10790

cre 10791

cim 10792

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 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879

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 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-mpt 4050 df-id 4276 df-po 4279 df-iso 4280 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-2 8924 df-cj 10793 df-re 10794 df-im 10795

This theorem is referenced by: remul 10823 immul 10830 cjmul 10836

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