mulgdirlem - Intuitionistic Logic Explorer

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Theorem mulgdirlem 12902

Description: Lemma for mulgdir 12903. (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 1000	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
2	1	grpmndd 12779	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
3		simprl 529	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
4		simprr 531	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
5		simpl23 1077	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
6		mulgnndir.b	. . . . . 6
7		mulgnndir.t	. . . . . 6 ._g
8		mulgnndir.p	. . . . . 6
9	6, 7, 8	mulgnn0dir 12901	. . . . 5 $/\$ $/\$ $/\$
10	2, 3, 4, 5, 9	syl13anc 1240	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
11	10	anassrs 400	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
12		simpl1 1000	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
13		simp22 1031	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$
14	13	adantr 276	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
15		simpl23 1077	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
16		eqid 2177	. . . . . . . . . . 11
17	6, 7, 16	mulgneg 12890	. . . . . . . . . 10 $/\$ $/\$
18	12, 14, 15, 17	syl3anc 1238	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
19	18	oveq1d 5884	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
20	6, 7	mulgcl 12889	. . . . . . . . . 10 $/\$ $/\$
21	12, 14, 15, 20	syl3anc 1238	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
22		eqid 2177	. . . . . . . . . 10
23	6, 8, 22, 16	grplinv 12812	. . . . . . . . 9 $/\$
24	12, 21, 23	syl2anc 411	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
25	19, 24	eqtrd 2210	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
26	25	oveq2d 5885	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
27		simpl3 1002	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
28		nn0z 9262	. . . . . . . . 9
29	27, 28	syl 14	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
30	6, 7	mulgcl 12889	. . . . . . . 8 $/\$ $/\$
31	12, 29, 15, 30	syl3anc 1238	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
32	6, 8, 22	grprid 12797	. . . . . . 7 $/\$
33	12, 31, 32	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
34	26, 33	eqtrd 2210	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
35		nn0z 9262	. . . . . . . . 9
36	35	ad2antll 491	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
37	6, 7	mulgcl 12889	. . . . . . . 8 $/\$ $/\$
38	12, 36, 15, 37	syl3anc 1238	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
39	6, 8	grpass 12776	. . . . . . 7 $/\$ $/\$ $/\$
40	12, 31, 38, 21, 39	syl13anc 1240	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
41	12	grpmndd 12779	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
42		simprr 531	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
43	6, 7, 8	mulgnn0dir 12901	. . . . . . . . 9 $/\$ $/\$ $/\$
44	41, 27, 42, 15, 43	syl13anc 1240	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
45		simp21 1030	. . . . . . . . . . . . . 14 $/\$ $/\$ $/\$ $/\$
46	45	zcnd 9365	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
47	13	zcnd 9365	. . . . . . . . . . . . 13 $/\$ $/\$ $/\$ $/\$
48	46, 47	addcld 7967	. . . . . . . . . . . 12 $/\$ $/\$ $/\$ $/\$
49	48	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
50	47	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
51	49, 50	negsubd 8264	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
52	46	adantr 276	. . . . . . . . . . 11 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
53	52, 50	pncand 8259	. . . . . . . . . 10 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
54	51, 53	eqtrd 2210	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
55	54	oveq1d 5884	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
56	44, 55	eqtr3d 2212	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
57	56	oveq1d 5884	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
58	40, 57	eqtr3d 2212	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
59	34, 58	eqtr3d 2212	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
60	59	anassrs 400	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $/\$
61		elznn0 9257	. . . . . 6 $/\$ $\/$
62	61	simprbi 275	. . . . 5 $\/$
63	13, 62	syl 14	. . . 4 $/\$ $/\$ $/\$ $/\$ $\/$
64	63	adantr 276	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$ $\/$
65	11, 60, 64	mpjaodan 798	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
66		simpl1 1000	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
67	45	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
68		simpl23 1077	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
69	6, 7	mulgcl 12889	. . . . 5 $/\$ $/\$
70	66, 67, 68, 69	syl3anc 1238	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
71	67	znegcld 9366	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
72	6, 7	mulgcl 12889	. . . . 5 $/\$ $/\$
73	66, 71, 68, 72	syl3anc 1238	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
74	28	3ad2ant3 1020	. . . . . 6 $/\$ $/\$ $/\$ $/\$
75	74	adantr 276	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
76	66, 75, 68, 30	syl3anc 1238	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
77	6, 8	grpass 12776	. . . 4 $/\$ $/\$ $/\$
78	66, 70, 73, 76, 77	syl13anc 1240	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
79	6, 7, 16	mulgneg 12890	. . . . . . . 8 $/\$ $/\$
80	66, 67, 68, 79	syl3anc 1238	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
81	80	oveq2d 5885	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
82	6, 8, 22, 16	grprinv 12813	. . . . . . 7 $/\$
83	66, 70, 82	syl2anc 411	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
84	81, 83	eqtrd 2210	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
85	84	oveq1d 5884	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
86	6, 8, 22	grplid 12796	. . . . 5 $/\$
87	66, 76, 86	syl2anc 411	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
88	85, 87	eqtrd 2210	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
89	66	grpmndd 12779	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
90		simpr 110	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
91		simpl3 1002	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
92	6, 7, 8	mulgnn0dir 12901	. . . . . 6 $/\$ $/\$ $/\$
93	89, 90, 91, 68, 92	syl13anc 1240	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
94	46	adantr 276	. . . . . . . . 9 $/\$ $/\$ $/\$ $/\$ $/\$
95	94	negcld 8245	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
96	48	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
97	95, 96	addcomd 8098	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
98	96, 94	negsubd 8264	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
99	47	adantr 276	. . . . . . . 8 $/\$ $/\$ $/\$ $/\$ $/\$
100	94, 99	pncan2d 8260	. . . . . . 7 $/\$ $/\$ $/\$ $/\$ $/\$
101	97, 98, 100	3eqtrd 2214	. . . . . 6 $/\$ $/\$ $/\$ $/\$ $/\$
102	101	oveq1d 5884	. . . . 5 $/\$ $/\$ $/\$ $/\$ $/\$
103	93, 102	eqtr3d 2212	. . . 4 $/\$ $/\$ $/\$ $/\$ $/\$
104	103	oveq2d 5885	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
105	78, 88, 104	3eqtr3d 2218	. 2 $/\$ $/\$ $/\$ $/\$ $/\$
106		elznn0 9257	. . . 4 $/\$ $\/$
107	106	simprbi 275	. . 3 $\/$
108	45, 107	syl 14	. 2 $/\$ $/\$ $/\$ $/\$ $\/$
109	65, 105, 108	mpjaodan 798	1 $/\$ $/\$ $/\$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104 $\/$ wo 708 $/\$ w3a 978

wceq 1353

wcel 2148

cfv 5212 (class class class)co 5869

cc 7800

cr 7801

caddc 7805

cmin 8118

cneg 8119

cn0 9165

cz 9242

cbs 12445

cplusg 12518

c0g 12653

cmnd 12709

cgrp 12767

cminusg 12768 ._gcmg 12872

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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-addcom 7902 ax-addass 7904 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-0id 7910 ax-rnegex 7911 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-ltadd 7918

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-inn 8909 df-2 8967 df-n0 9166 df-z 9243 df-uz 9518 df-fz 9996 df-seqfrec 10432 df-ndx 12448 df-slot 12449 df-base 12451 df-plusg 12531 df-0g 12655 df-mgm 12667 df-sgrp 12700 df-mnd 12710 df-grp 12770 df-minusg 12771 df-mulg 12873

This theorem is referenced by: mulgdir 12903

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