mulgdirlem - Intuitionistic Logic Explorer

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Theorem mulgdirlem 13739

Description: Lemma for mulgdir 13740. (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 1026	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2	1	grpmndd 13595	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
3		simprl 531	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 533	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5		simpl23 1103	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		mulgnndir.b	. . . . . 6
7		mulgnndir.t	. . . . . 6 ._g
8		mulgnndir.p	. . . . . 6
9	6, 7, 8	mulgnn0dir 13738	. . . . 5 $/\$ $/\$ $/\$
10	2, 3, 4, 5, 9	syl13anc 1275	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11	10	anassrs 400	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simpl1 1026	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simp22 1057	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
14	13	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl23 1103	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		eqid 2231	. . . . . . . . . . 11
17	6, 7, 16	mulgneg 13726	. . . . . . . . . 10 $/\$ $/\$
18	12, 14, 15, 17	syl3anc 1273	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	18	oveq1d 6032	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	6, 7	mulgcl 13725	. . . . . . . . . 10 $/\$ $/\$
21	12, 14, 15, 20	syl3anc 1273	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eqid 2231	. . . . . . . . . 10
23	6, 8, 22, 16	grplinv 13632	. . . . . . . . 9 $/\$
24	12, 21, 23	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 24	eqtrd 2264	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 6033	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		simpl3 1028	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		nn0z 9498	. . . . . . . . 9
29	27, 28	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	6, 7	mulgcl 13725	. . . . . . . 8 $/\$ $/\$
31	12, 29, 15, 30	syl3anc 1273	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	6, 8, 22	grprid 13614	. . . . . . 7 $/\$
33	12, 31, 32	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	26, 33	eqtrd 2264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		nn0z 9498	. . . . . . . . 9
36	35	ad2antll 491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	6, 7	mulgcl 13725	. . . . . . . 8 $/\$ $/\$
38	12, 36, 15, 37	syl3anc 1273	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	6, 8	grpass 13591	. . . . . . 7 $/\$ $/\$ $/\$
40	12, 31, 38, 21, 39	syl13anc 1275	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	12	grpmndd 13595	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		simprr 533	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	6, 7, 8	mulgnn0dir 13738	. . . . . . . . 9 $/\$ $/\$ $/\$
44	41, 27, 42, 15, 43	syl13anc 1275	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp21 1056	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
46	45	zcnd 9602	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
47	13	zcnd 9602	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
48	46, 47	addcld 8198	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
49	48	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	47	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	49, 50	negsubd 8495	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	46	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	52, 50	pncand 8490	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	51, 53	eqtrd 2264	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	54	oveq1d 6032	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	44, 55	eqtr3d 2266	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	56	oveq1d 6032	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	40, 57	eqtr3d 2266	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	34, 58	eqtr3d 2266	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	59	anassrs 400	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61		elznn0 9493	. . . . . 6 $/\$ $\/$
62	61	simprbi 275	. . . . 5 $\/$
63	13, 62	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$
64	63	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
65	11, 60, 64	mpjaodan 805	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
66		simpl1 1026	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
67	45	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
68		simpl23 1103	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
69	6, 7	mulgcl 13725	. . . . 5 $/\$ $/\$
70	66, 67, 68, 69	syl3anc 1273	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
71	67	znegcld 9603	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
72	6, 7	mulgcl 13725	. . . . 5 $/\$ $/\$
73	66, 71, 68, 72	syl3anc 1273	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	28	3ad2ant3 1046	. . . . . 6 $/\$ $/\$ $/\$ $/\$
75	74	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
76	66, 75, 68, 30	syl3anc 1273	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	6, 8	grpass 13591	. . . 4 $/\$ $/\$ $/\$
78	66, 70, 73, 76, 77	syl13anc 1275	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
79	6, 7, 16	mulgneg 13726	. . . . . . . 8 $/\$ $/\$
80	66, 67, 68, 79	syl3anc 1273	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
81	80	oveq2d 6033	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	6, 8, 22, 16	grprinv 13633	. . . . . . 7 $/\$
83	66, 70, 82	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
84	81, 83	eqtrd 2264	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
85	84	oveq1d 6032	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
86	6, 8, 22	grplid 13613	. . . . 5 $/\$
87	66, 76, 86	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
88	85, 87	eqtrd 2264	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
89	66	grpmndd 13595	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
90		simpr 110	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
91		simpl3 1028	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
92	6, 7, 8	mulgnn0dir 13738	. . . . . 6 $/\$ $/\$ $/\$
93	89, 90, 91, 68, 92	syl13anc 1275	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
94	46	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
95	94	negcld 8476	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
96	48	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	95, 96	addcomd 8329	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
98	96, 94	negsubd 8495	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
99	47	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	94, 99	pncan2d 8491	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	97, 98, 100	3eqtrd 2268	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	101	oveq1d 6032	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	93, 102	eqtr3d 2266	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	oveq2d 6033	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
105	78, 88, 104	3eqtr3d 2272	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
106		elznn0 9493	. . . 4 $/\$ $\/$
107	106	simprbi 275	. . 3 $\/$
108	45, 107	syl 14	. 2 $/\$ $/\$ $/\$ $/\$ $\/$
109	65, 105, 108	mpjaodan 805	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 715 $/\$ w3a 1004

wceq 1397

wcel 2202

cfv 5326 (class class class)co 6017

cc 8029

cr 8030

caddc 8034

cmin 8349

cneg 8350

cn0 9401

cz 9478

cbs 13081

cplusg 13159

c0g 13338

cmnd 13498

cgrp 13582

cminusg 13583 ._gcmg 13705

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8122 ax-resscn 8123 ax-1cn 8124 ax-1re 8125 ax-icn 8126 ax-addcl 8127 ax-addrcl 8128 ax-mulcl 8129 ax-addcom 8131 ax-addass 8133 ax-distr 8135 ax-i2m1 8136 ax-0lt1 8137 ax-0id 8139 ax-rnegex 8140 ax-cnre 8142 ax-pre-ltirr 8143 ax-pre-ltwlin 8144 ax-pre-lttrn 8145 ax-pre-ltadd 8147

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rmo 2518 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5970 df-ov 6020 df-oprab 6021 df-mpo 6022 df-1st 6302 df-2nd 6303 df-recs 6470 df-frec 6556 df-pnf 8215 df-mnf 8216 df-xr 8217 df-ltxr 8218 df-le 8219 df-sub 8351 df-neg 8352 df-inn 9143 df-2 9201 df-n0 9402 df-z 9479 df-uz 9755 df-fz 10243 df-seqfrec 10709 df-ndx 13084 df-slot 13085 df-base 13087 df-plusg 13172 df-0g 13340 df-mgm 13438 df-sgrp 13484 df-mnd 13499 df-grp 13585 df-minusg 13586 df-mulg 13706

This theorem is referenced by: mulgdir 13740

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