Theorem mndodconglem 17960

Description: Lemma for mndodcong 17961. (Contributed by Mario Carneiro, 23-Sep-2015.)

Hypotheses

Ref	Expression
odcl.1	|- X = ( Base ` G )
odcl.2	|- O = ( od ` G )
odid.3	|- .x. = (.g ` G )
odid.4	|- .0. = ( 0g ` G )
mndodconglem.1	|- ( ph -> G e. Mnd )
mndodconglem.2	|- ( ph -> A e. X )
mndodconglem.3	|- ( ph -> ( O ` A ) e. NN )
mndodconglem.4	|- ( ph -> M e. NN0 )
mndodconglem.5	|- ( ph -> N e. NN0 )
mndodconglem.6	|- ( ph -> M < ( O ` A ) )
mndodconglem.7	|- ( ph -> N < ( O ` A ) )
mndodconglem.8	|- ( ph -> ( M .x. A ) = ( N .x. A ) )

Assertion

Ref	Expression
mndodconglem	|- ( ( ph /\ M <_ N ) -> M = N )

Proof of Theorem mndodconglem

Step	Hyp	Ref	Expression
1		mndodconglem.2	. . . . . . 7 |- ( ph -> A e. X )
2		mndodconglem.3	. . . . . . . . . . 11 |- ( ph -> ( O ` A ) e. NN )
3	2	nnred 11035	. . . . . . . . . 10 |- ( ph -> ( O ` A ) e. RR )
4	3	recnd 10068	. . . . . . . . 9 |- ( ph -> ( O ` A ) e. CC )
5		mndodconglem.4	. . . . . . . . . . 11 |- ( ph -> M e. NN0 )
6	5	nn0red 11352	. . . . . . . . . 10 |- ( ph -> M e. RR )
7	6	recnd 10068	. . . . . . . . 9 |- ( ph -> M e. CC )
8		mndodconglem.5	. . . . . . . . . . 11 |- ( ph -> N e. NN0 )
9	8	nn0red 11352	. . . . . . . . . 10 |- ( ph -> N e. RR )
10	9	recnd 10068	. . . . . . . . 9 |- ( ph -> N e. CC )
11	4, 7, 10	addsubassd 10412	. . . . . . . 8 |- ( ph -> ( ( ( O ` A ) + M ) - N ) = ( ( O ` A ) + ( M - N ) ) )
12	2	nnzd 11481	. . . . . . . . . . . 12 |- ( ph -> ( O ` A ) e. ZZ )
13	5	nn0zd 11480	. . . . . . . . . . . 12 |- ( ph -> M e. ZZ )
14	12, 13	zaddcld 11486	. . . . . . . . . . 11 |- ( ph -> ( ( O ` A ) + M ) e. ZZ )
15	14	zred 11482	. . . . . . . . . 10 |- ( ph -> ( ( O ` A ) + M ) e. RR )
16		mndodconglem.7	. . . . . . . . . 10 |- ( ph -> N < ( O ` A ) )
17		nn0addge1 11339	. . . . . . . . . . 11 |- ( ( ( O ` A ) e. RR /\ M e. NN0 ) -> ( O ` A ) <_ ( ( O ` A ) + M ) )
18	3, 5, 17	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( O ` A ) <_ ( ( O ` A ) + M ) )
19	9, 3, 15, 16, 18	ltletrd 10197	. . . . . . . . 9 |- ( ph -> N < ( ( O ` A ) + M ) )
20	8	nn0zd 11480	. . . . . . . . . 10 |- ( ph -> N e. ZZ )
21		znnsub 11423	. . . . . . . . . 10 |- ( ( N e. ZZ /\ ( ( O ` A ) + M ) e. ZZ ) -> ( N < ( ( O ` A ) + M ) <-> ( ( ( O ` A ) + M ) - N ) e. NN ) )
22	20, 14, 21	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( N < ( ( O ` A ) + M ) <-> ( ( ( O ` A ) + M ) - N ) e. NN ) )
23	19, 22	mpbid 222	. . . . . . . 8 |- ( ph -> ( ( ( O ` A ) + M ) - N ) e. NN )
24	11, 23	eqeltrrd 2702	. . . . . . 7 |- ( ph -> ( ( O ` A ) + ( M - N ) ) e. NN )
25	4, 7, 10	addsub12d 10415	. . . . . . . . 9 |- ( ph -> ( ( O ` A ) + ( M - N ) ) = ( M + ( ( O ` A ) - N ) ) )
26	25	oveq1d 6665	. . . . . . . 8 |- ( ph -> ( ( ( O ` A ) + ( M - N ) ) .x. A ) = ( ( M + ( ( O ` A ) - N ) ) .x. A ) )
27		mndodconglem.8	. . . . . . . . . . 11 |- ( ph -> ( M .x. A ) = ( N .x. A ) )
28	27	oveq1d 6665	. . . . . . . . . 10 |- ( ph -> ( ( M .x. A ) ( +g ` G ) ( ( ( O ` A ) - N ) .x. A ) ) = ( ( N .x. A ) ( +g ` G ) ( ( ( O ` A ) - N ) .x. A ) ) )
29		mndodconglem.1	. . . . . . . . . . 11 |- ( ph -> G e. Mnd )
30		znnsub 11423	. . . . . . . . . . . . . 14 |- ( ( N e. ZZ /\ ( O ` A ) e. ZZ ) -> ( N < ( O ` A ) <-> ( ( O ` A ) - N ) e. NN ) )
31	20, 12, 30	syl2anc 693	. . . . . . . . . . . . 13 |- ( ph -> ( N < ( O ` A ) <-> ( ( O ` A ) - N ) e. NN ) )
32	16, 31	mpbid 222	. . . . . . . . . . . 12 |- ( ph -> ( ( O ` A ) - N ) e. NN )
33	32	nnnn0d 11351	. . . . . . . . . . 11 |- ( ph -> ( ( O ` A ) - N ) e. NN0 )
34		odcl.1	. . . . . . . . . . . 12 |- X = ( Base ` G )
35		odid.3	. . . . . . . . . . . 12 |- .x. = (.g ` G )
36		eqid 2622	. . . . . . . . . . . 12 |- ( +g ` G ) = ( +g ` G )
37	34, 35, 36	mulgnn0dir 17571	. . . . . . . . . . 11 |- ( ( G e. Mnd /\ ( M e. NN0 /\ ( ( O ` A ) - N ) e. NN0 /\ A e. X ) ) -> ( ( M + ( ( O ` A ) - N ) ) .x. A ) = ( ( M .x. A ) ( +g ` G ) ( ( ( O ` A ) - N ) .x. A ) ) )
38	29, 5, 33, 1, 37	syl13anc 1328	. . . . . . . . . 10 |- ( ph -> ( ( M + ( ( O ` A ) - N ) ) .x. A ) = ( ( M .x. A ) ( +g ` G ) ( ( ( O ` A ) - N ) .x. A ) ) )
39	34, 35, 36	mulgnn0dir 17571	. . . . . . . . . . 11 |- ( ( G e. Mnd /\ ( N e. NN0 /\ ( ( O ` A ) - N ) e. NN0 /\ A e. X ) ) -> ( ( N + ( ( O ` A ) - N ) ) .x. A ) = ( ( N .x. A ) ( +g ` G ) ( ( ( O ` A ) - N ) .x. A ) ) )
40	29, 8, 33, 1, 39	syl13anc 1328	. . . . . . . . . 10 |- ( ph -> ( ( N + ( ( O ` A ) - N ) ) .x. A ) = ( ( N .x. A ) ( +g ` G ) ( ( ( O ` A ) - N ) .x. A ) ) )
41	28, 38, 40	3eqtr4d 2666	. . . . . . . . 9 |- ( ph -> ( ( M + ( ( O ` A ) - N ) ) .x. A ) = ( ( N + ( ( O ` A ) - N ) ) .x. A ) )
42	10, 4	pncan3d 10395	. . . . . . . . . . 11 |- ( ph -> ( N + ( ( O ` A ) - N ) ) = ( O ` A ) )
43	42	oveq1d 6665	. . . . . . . . . 10 |- ( ph -> ( ( N + ( ( O ` A ) - N ) ) .x. A ) = ( ( O ` A ) .x. A ) )
44		odcl.2	. . . . . . . . . . . 12 |- O = ( od ` G )
45		odid.4	. . . . . . . . . . . 12 |- .0. = ( 0g ` G )
46	34, 44, 35, 45	odid 17957	. . . . . . . . . . 11 |- ( A e. X -> ( ( O ` A ) .x. A ) = .0. )
47	1, 46	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( O ` A ) .x. A ) = .0. )
48	43, 47	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( N + ( ( O ` A ) - N ) ) .x. A ) = .0. )
49	41, 48	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( ( M + ( ( O ` A ) - N ) ) .x. A ) = .0. )
50	26, 49	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( ( O ` A ) + ( M - N ) ) .x. A ) = .0. )
51	34, 44, 35, 45	odlem2 17958	. . . . . . 7 |- ( ( A e. X /\ ( ( O ` A ) + ( M - N ) ) e. NN /\ ( ( ( O ` A ) + ( M - N ) ) .x. A ) = .0. ) -> ( O ` A ) e. ( 1 ... ( ( O ` A ) + ( M - N ) ) ) )
52	1, 24, 50, 51	syl3anc 1326	. . . . . 6 |- ( ph -> ( O ` A ) e. ( 1 ... ( ( O ` A ) + ( M - N ) ) ) )
53		elfzle2 12345	. . . . . 6 |- ( ( O ` A ) e. ( 1 ... ( ( O ` A ) + ( M - N ) ) ) -> ( O ` A ) <_ ( ( O ` A ) + ( M - N ) ) )
54	52, 53	syl 17	. . . . 5 |- ( ph -> ( O ` A ) <_ ( ( O ` A ) + ( M - N ) ) )
55	13, 20	zsubcld 11487	. . . . . . 7 |- ( ph -> ( M - N ) e. ZZ )
56	55	zred 11482	. . . . . 6 |- ( ph -> ( M - N ) e. RR )
57	3, 56	addge01d 10615	. . . . 5 |- ( ph -> ( 0 <_ ( M - N ) <-> ( O ` A ) <_ ( ( O ` A ) + ( M - N ) ) ) )
58	54, 57	mpbird 247	. . . 4 |- ( ph -> 0 <_ ( M - N ) )
59	6, 9	subge0d 10617	. . . 4 |- ( ph -> ( 0 <_ ( M - N ) <-> N <_ M ) )
60	58, 59	mpbid 222	. . 3 |- ( ph -> N <_ M )
61	6, 9	letri3d 10179	. . . 4 |- ( ph -> ( M = N <-> ( M <_ N /\ N <_ M ) ) )
62	61	biimprd 238	. . 3 |- ( ph -> ( ( M <_ N /\ N <_ M ) -> M = N ) )
63	60, 62	mpan2d 710	. 2 |- ( ph -> ( M <_ N -> M = N ) )
64	63	imp 445	1 |- ( ( ph /\ M <_ N ) -> M = N )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326   Basecbs 15857   +g cplusg 15941   0gc0g 16100   Mndcmnd 17294  ._gcmg 17540   odcod 17944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mulg 17541  df-od 17948

This theorem is referenced by:  mndodcong  17961