mulgdirlem - Intuitionistic Logic Explorer

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Theorem mulgdirlem 13283

Description: Lemma for mulgdir 13284. (Contributed by Mario Carneiro, 13-Dec-2014.)

**Hypotheses**
Ref	Expression
mulgnndir.b
mulgnndir.t	._g
mulgnndir.p

**Assertion**
Ref	Expression
mulgdirlem	$/\$ $/\$ $/\$ $/\$

**Proof of Theorem mulgdirlem**
Step	Hyp	Ref	Expression
1		simpl1 1002	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2	1	grpmndd 13145	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
3		simprl 529	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 531	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5		simpl23 1079	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		mulgnndir.b	. . . . . 6
7		mulgnndir.t	. . . . . 6 ._g
8		mulgnndir.p	. . . . . 6
9	6, 7, 8	mulgnn0dir 13282	. . . . 5 $/\$ $/\$ $/\$
10	2, 3, 4, 5, 9	syl13anc 1251	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11	10	anassrs 400	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simpl1 1002	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simp22 1033	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
14	13	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl23 1079	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		eqid 2196	. . . . . . . . . . 11
17	6, 7, 16	mulgneg 13270	. . . . . . . . . 10 $/\$ $/\$
18	12, 14, 15, 17	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	18	oveq1d 5937	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	6, 7	mulgcl 13269	. . . . . . . . . 10 $/\$ $/\$
21	12, 14, 15, 20	syl3anc 1249	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eqid 2196	. . . . . . . . . 10
23	6, 8, 22, 16	grplinv 13182	. . . . . . . . 9 $/\$
24	12, 21, 23	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 24	eqtrd 2229	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5938	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		simpl3 1004	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		nn0z 9346	. . . . . . . . 9
29	27, 28	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	6, 7	mulgcl 13269	. . . . . . . 8 $/\$ $/\$
31	12, 29, 15, 30	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	6, 8, 22	grprid 13164	. . . . . . 7 $/\$
33	12, 31, 32	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	26, 33	eqtrd 2229	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		nn0z 9346	. . . . . . . . 9
36	35	ad2antll 491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	6, 7	mulgcl 13269	. . . . . . . 8 $/\$ $/\$
38	12, 36, 15, 37	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	6, 8	grpass 13141	. . . . . . 7 $/\$ $/\$ $/\$
40	12, 31, 38, 21, 39	syl13anc 1251	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	12	grpmndd 13145	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	6, 7, 8	mulgnn0dir 13282	. . . . . . . . 9 $/\$ $/\$ $/\$
44	41, 27, 42, 15, 43	syl13anc 1251	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp21 1032	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
46	45	zcnd 9449	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
47	13	zcnd 9449	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
48	46, 47	addcld 8046	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
49	48	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	47	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	49, 50	negsubd 8343	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	46	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	52, 50	pncand 8338	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	51, 53	eqtrd 2229	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	54	oveq1d 5937	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	44, 55	eqtr3d 2231	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	56	oveq1d 5937	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	40, 57	eqtr3d 2231	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	34, 58	eqtr3d 2231	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	59	anassrs 400	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61		elznn0 9341	. . . . . 6 $/\$ $\/$
62	61	simprbi 275	. . . . 5 $\/$
63	13, 62	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$
64	63	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
65	11, 60, 64	mpjaodan 799	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
66		simpl1 1002	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
67	45	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
68		simpl23 1079	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
69	6, 7	mulgcl 13269	. . . . 5 $/\$ $/\$
70	66, 67, 68, 69	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
71	67	znegcld 9450	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
72	6, 7	mulgcl 13269	. . . . 5 $/\$ $/\$
73	66, 71, 68, 72	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	28	3ad2ant3 1022	. . . . . 6 $/\$ $/\$ $/\$ $/\$
75	74	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
76	66, 75, 68, 30	syl3anc 1249	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	6, 8	grpass 13141	. . . 4 $/\$ $/\$ $/\$
78	66, 70, 73, 76, 77	syl13anc 1251	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
79	6, 7, 16	mulgneg 13270	. . . . . . . 8 $/\$ $/\$
80	66, 67, 68, 79	syl3anc 1249	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
81	80	oveq2d 5938	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	6, 8, 22, 16	grprinv 13183	. . . . . . 7 $/\$
83	66, 70, 82	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
84	81, 83	eqtrd 2229	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
85	84	oveq1d 5937	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
86	6, 8, 22	grplid 13163	. . . . 5 $/\$
87	66, 76, 86	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
88	85, 87	eqtrd 2229	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
89	66	grpmndd 13145	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
90		simpr 110	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
91		simpl3 1004	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
92	6, 7, 8	mulgnn0dir 13282	. . . . . 6 $/\$ $/\$ $/\$
93	89, 90, 91, 68, 92	syl13anc 1251	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
94	46	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
95	94	negcld 8324	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
96	48	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	95, 96	addcomd 8177	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
98	96, 94	negsubd 8343	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
99	47	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	94, 99	pncan2d 8339	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	97, 98, 100	3eqtrd 2233	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	101	oveq1d 5937	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	93, 102	eqtr3d 2231	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	oveq2d 5938	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
105	78, 88, 104	3eqtr3d 2237	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
106		elznn0 9341	. . . 4 $/\$ $\/$
107	106	simprbi 275	. . 3 $\/$
108	45, 107	syl 14	. 2 $/\$ $/\$ $/\$ $/\$ $\/$
109	65, 105, 108	mpjaodan 799	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 709 $/\$ w3a 980

wceq 1364

wcel 2167

cfv 5258 (class class class)co 5922

cc 7877

cr 7878

caddc 7882

cmin 8197

cneg 8198

cn0 9249

cz 9326

cbs 12678

cplusg 12755

c0g 12927

cmnd 13057

cgrp 13132

cminusg 13133 ._gcmg 13249

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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-coll 4148 ax-sep 4151 ax-nul 4159 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-iinf 4624 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-addcom 7979 ax-addass 7981 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-0id 7987 ax-rnegex 7988 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-ltadd 7995

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-nul 3451 df-if 3562 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-tr 4132 df-id 4328 df-iord 4401 df-on 4403 df-ilim 4404 df-suc 4406 df-iom 4627 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-f1 5263 df-fo 5264 df-f1o 5265 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-recs 6363 df-frec 6449 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-inn 8991 df-2 9049 df-n0 9250 df-z 9327 df-uz 9602 df-fz 10084 df-seqfrec 10540 df-ndx 12681 df-slot 12682 df-base 12684 df-plusg 12768 df-0g 12929 df-mgm 12999 df-sgrp 13045 df-mnd 13058 df-grp 13135 df-minusg 13136 df-mulg 13250

This theorem is referenced by: mulgdir 13284

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