remullem - Intuitionistic Logic Explorer

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Theorem remullem 11614

Description: Lemma for remul 11615, immul 11622, and cjmul 11628. (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 11602	. . . . . 6
2		replim 11602	. . . . . 6
3	1, 2	oveqan12d 6094	. . . . 5 $/\$
4		recl 11596	. . . . . . . . 9
5	4	adantr 276	. . . . . . . 8 $/\$
6	5	recnd 8344	. . . . . . 7 $/\$
7		ax-icn 8264	. . . . . . . 8
8		imcl 11597	. . . . . . . . . 10
9	8	adantr 276	. . . . . . . . 9 $/\$
10	9	recnd 8344	. . . . . . . 8 $/\$
11		mulcl 8296	. . . . . . . 8 $/\$
12	7, 10, 11	sylancr 418	. . . . . . 7 $/\$
13	6, 12	addcld 8335	. . . . . 6 $/\$
14		recl 11596	. . . . . . . 8
15	14	adantl 277	. . . . . . 7 $/\$
16	15	recnd 8344	. . . . . 6 $/\$
17		imcl 11597	. . . . . . . . 9
18	17	adantl 277	. . . . . . . 8 $/\$
19	18	recnd 8344	. . . . . . 7 $/\$
20		mulcl 8296	. . . . . . 7 $/\$
21	7, 19, 20	sylancr 418	. . . . . 6 $/\$
22	13, 16, 21	adddid 8340	. . . . 5 $/\$
23	6, 12, 16	adddird 8341	. . . . . . 7 $/\$
24	6, 12, 21	adddird 8341	. . . . . . 7 $/\$
25	23, 24	oveq12d 6093	. . . . . 6 $/\$
26	5, 15	remulcld 8346	. . . . . . . 8 $/\$
27	26	recnd 8344	. . . . . . 7 $/\$
28	12, 21	mulcld 8336	. . . . . . 7 $/\$
29	12, 16	mulcld 8336	. . . . . . 7 $/\$
30	6, 21	mulcld 8336	. . . . . . 7 $/\$
31	27, 28, 29, 30	add42d 8486	. . . . . 6 $/\$
32	7	a1i 9	. . . . . . . . . . 11 $/\$
33	32, 10, 32, 19	mul4d 8471	. . . . . . . . . 10 $/\$
34		ixi 8901	. . . . . . . . . . . 12
35	34	oveq1i 6085	. . . . . . . . . . 11
36	9, 18	remulcld 8346	. . . . . . . . . . . . 13 $/\$
37	36	recnd 8344	. . . . . . . . . . . 12 $/\$
38	37	mulm1d 8727	. . . . . . . . . . 11 $/\$
39	35, 38	eqtrid 2283	. . . . . . . . . 10 $/\$
40	33, 39	eqtrd 2271	. . . . . . . . 9 $/\$
41	40	oveq2d 6091	. . . . . . . 8 $/\$
42	27, 37	negsubd 8633	. . . . . . . 8 $/\$
43	41, 42	eqtrd 2271	. . . . . . 7 $/\$
44	9, 15	remulcld 8346	. . . . . . . . . . 11 $/\$
45	44	recnd 8344	. . . . . . . . . 10 $/\$
46		mulcl 8296	. . . . . . . . . 10 $/\$
47	7, 45, 46	sylancr 418	. . . . . . . . 9 $/\$
48	5, 18	remulcld 8346	. . . . . . . . . . 11 $/\$
49	48	recnd 8344	. . . . . . . . . 10 $/\$
50		mulcl 8296	. . . . . . . . . 10 $/\$
51	7, 49, 50	sylancr 418	. . . . . . . . 9 $/\$
52	47, 51	addcomd 8467	. . . . . . . 8 $/\$
53	32, 10, 16	mulassd 8339	. . . . . . . . 9 $/\$
54	6, 32, 19	mul12d 8468	. . . . . . . . 9 $/\$
55	53, 54	oveq12d 6093	. . . . . . . 8 $/\$
56	32, 49, 45	adddid 8340	. . . . . . . 8 $/\$
57	52, 55, 56	3eqtr4d 2281	. . . . . . 7 $/\$
58	43, 57	oveq12d 6093	. . . . . 6 $/\$
59	25, 31, 58	3eqtr2d 2277	. . . . 5 $/\$
60	3, 22, 59	3eqtrd 2275	. . . 4 $/\$
61	60	fveq2d 5694	. . 3 $/\$
62	26, 36	resubcld 8698	. . . 4 $/\$
63	48, 44	readdcld 8345	. . . 4 $/\$
64		crre 11600	. . . 4 $/\$
65	62, 63, 64	syl2anc 415	. . 3 $/\$
66	61, 65	eqtrd 2271	. 2 $/\$
67	60	fveq2d 5694	. . 3 $/\$
68		crim 11601	. . . 4 $/\$
69	62, 63, 68	syl2anc 415	. . 3 $/\$
70	67, 69	eqtrd 2271	. 2 $/\$
71		mulcl 8296	. . . 4 $/\$
72		remim 11603	. . . 4
73	71, 72	syl 14	. . 3 $/\$
74		remim 11603	. . . . 5
75		remim 11603	. . . . 5
76	74, 75	oveqan12d 6094	. . . 4 $/\$
77	16, 21	subcld 8627	. . . . 5 $/\$
78	6, 12, 77	subdird 8732	. . . 4 $/\$
79	27, 30, 29, 28	subadd4d 8675	. . . . 5 $/\$
80	6, 16, 21	subdid 8731	. . . . . 6 $/\$
81	12, 16, 21	subdid 8731	. . . . . 6 $/\$
82	80, 81	oveq12d 6093	. . . . 5 $/\$
83	65, 61, 43	3eqtr4d 2281	. . . . . 6 $/\$
84	70	oveq2d 6091	. . . . . . 7 $/\$
85	54, 53	oveq12d 6093	. . . . . . 7 $/\$
86	56, 84, 85	3eqtr4d 2281	. . . . . 6 $/\$
87	83, 86	oveq12d 6093	. . . . 5 $/\$
88	79, 82, 87	3eqtr4d 2281	. . . 4 $/\$
89	76, 78, 88	3eqtrd 2275	. . 3 $/\$
90	73, 89	eqtr4d 2274	. 2 $/\$
91	66, 70, 90	3jca 1208	1 $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $/\$ w3a 1009

wceq 1402

wcel 2209

cfv 5372 (class class class)co 6075

cc 8167

cr 8168

c1 8170

ci 8171

caddc 8172

cmul 8174

cmin 8487

cneg 8488

ccj 11582

cre 11583

cim 11584

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 ax-pre-mulext 8287

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rmo 2536 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-mpt 4189 df-id 4433 df-po 4436 df-iso 4437 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-rn 4780 df-res 4781 df-ima 4782 df-iota 5332 df-fun 5374 df-fn 5375 df-f 5376 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 df-div 8993 df-2 9342 df-cj 11585 df-re 11586 df-im 11587

This theorem is referenced by: remul 11615 immul 11622 cjmul 11628

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