Theorem fta1blem 23928

Description: Lemma for fta1b 23929. (Contributed by Mario Carneiro, 14-Jun-2015.)

Hypotheses

Ref	Expression
fta1b.p	|- P = (Poly1 ` R )
fta1b.b	|- B = ( Base ` P )
fta1b.d	|- D = ( deg1 ` R )
fta1b.o	|- O = (eval1 ` R )
fta1b.w	|- W = ( 0g ` R )
fta1b.z	|- .0. = ( 0g ` P )
fta1blem.k	|- K = ( Base ` R )
fta1blem.t	|- .X. = ( .r ` R )
fta1blem.x	|- X = (var1 ` R )
fta1blem.s	|- .x. = ( .s ` P )
fta1blem.1	|- ( ph -> R e. CRing )
fta1blem.2	|- ( ph -> M e. K )
fta1blem.3	|- ( ph -> N e. K )
fta1blem.4	|- ( ph -> ( M .X. N ) = W )
fta1blem.5	|- ( ph -> M =/= W )
fta1blem.6	|- ( ph -> ( ( M .x. X ) e. ( B \ { .0. } ) -> ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <_ ( D ` ( M .x. X ) ) ) )

Assertion

Ref	Expression
fta1blem	|- ( ph -> N = W )

Proof of Theorem fta1blem

Step	Hyp	Ref	Expression
1		fta1blem.3	. . . 4 |- ( ph -> N e. K )
2		fta1b.o	. . . . . . 7 |- O = (eval1 ` R )
3		fta1b.p	. . . . . . 7 |- P = (Poly1 ` R )
4		fta1blem.k	. . . . . . 7 |- K = ( Base ` R )
5		fta1b.b	. . . . . . 7 |- B = ( Base ` P )
6		fta1blem.1	. . . . . . 7 |- ( ph -> R e. CRing )
7		fta1blem.x	. . . . . . . 8 |- X = (var1 ` R )
8	2, 7, 4, 3, 5, 6, 1	evl1vard 19701	. . . . . . 7 |- ( ph -> ( X e. B /\ ( ( O ` X ) ` N ) = N ) )
9		fta1blem.2	. . . . . . 7 |- ( ph -> M e. K )
10		fta1blem.s	. . . . . . 7 |- .x. = ( .s ` P )
11		fta1blem.t	. . . . . . 7 |- .X. = ( .r ` R )
12	2, 3, 4, 5, 6, 1, 8, 9, 10, 11	evl1vsd 19708	. . . . . 6 |- ( ph -> ( ( M .x. X ) e. B /\ ( ( O ` ( M .x. X ) ) ` N ) = ( M .X. N ) ) )
13	12	simprd 479	. . . . 5 |- ( ph -> ( ( O ` ( M .x. X ) ) ` N ) = ( M .X. N ) )
14		fta1blem.4	. . . . 5 |- ( ph -> ( M .X. N ) = W )
15	13, 14	eqtrd 2656	. . . 4 |- ( ph -> ( ( O ` ( M .x. X ) ) ` N ) = W )
16		eqid 2622	. . . . . . 7 |- ( R ^s K ) = ( R ^s K )
17		eqid 2622	. . . . . . 7 |- ( Base ` ( R ^s K ) ) = ( Base ` ( R ^s K ) )
18		fvex 6201	. . . . . . . . 9 |- ( Base ` R ) e. _V
19	4, 18	eqeltri 2697	. . . . . . . 8 |- K e. _V
20	19	a1i 11	. . . . . . 7 |- ( ph -> K e. _V )
21	2, 3, 16, 4	evl1rhm 19696	. . . . . . . . . 10 |- ( R e. CRing -> O e. ( P RingHom ( R ^s K ) ) )
22	6, 21	syl 17	. . . . . . . . 9 |- ( ph -> O e. ( P RingHom ( R ^s K ) ) )
23	5, 17	rhmf 18726	. . . . . . . . 9 |- ( O e. ( P RingHom ( R ^s K ) ) -> O : B --> ( Base ` ( R ^s K ) ) )
24	22, 23	syl 17	. . . . . . . 8 |- ( ph -> O : B --> ( Base ` ( R ^s K ) ) )
25	12	simpld 475	. . . . . . . 8 |- ( ph -> ( M .x. X ) e. B )
26	24, 25	ffvelrnd 6360	. . . . . . 7 |- ( ph -> ( O ` ( M .x. X ) ) e. ( Base ` ( R ^s K ) ) )
27	16, 4, 17, 6, 20, 26	pwselbas 16149	. . . . . 6 |- ( ph -> ( O ` ( M .x. X ) ) : K --> K )
28	27	ffnd 6046	. . . . 5 |- ( ph -> ( O ` ( M .x. X ) ) Fn K )
29		fniniseg 6338	. . . . 5 |- ( ( O ` ( M .x. X ) ) Fn K -> ( N e. ( `' ( O ` ( M .x. X ) ) " { W } ) <-> ( N e. K /\ ( ( O ` ( M .x. X ) ) ` N ) = W ) ) )
30	28, 29	syl 17	. . . 4 |- ( ph -> ( N e. ( `' ( O ` ( M .x. X ) ) " { W } ) <-> ( N e. K /\ ( ( O ` ( M .x. X ) ) ` N ) = W ) ) )
31	1, 15, 30	mpbir2and 957	. . 3 |- ( ph -> N e. ( `' ( O ` ( M .x. X ) ) " { W } ) )
32		fvex 6201	. . . . . . . 8 |- ( O ` ( M .x. X ) ) e. _V
33	32	cnvex 7113	. . . . . . 7 |- `' ( O ` ( M .x. X ) ) e. _V
34	33	imaex 7104	. . . . . 6 |- ( `' ( O ` ( M .x. X ) ) " { W } ) e. _V
35	34	a1i 11	. . . . 5 |- ( ph -> ( `' ( O ` ( M .x. X ) ) " { W } ) e. _V )
36		1nn0 11308	. . . . . 6 |- 1 e. NN0
37	36	a1i 11	. . . . 5 |- ( ph -> 1 e. NN0 )
38		crngring 18558	. . . . . . . . . . . . 13 |- ( R e. CRing -> R e. Ring )
39	6, 38	syl 17	. . . . . . . . . . . 12 |- ( ph -> R e. Ring )
40	7, 3, 5	vr1cl 19587	. . . . . . . . . . . 12 |- ( R e. Ring -> X e. B )
41	39, 40	syl 17	. . . . . . . . . . 11 |- ( ph -> X e. B )
42		eqid 2622	. . . . . . . . . . . . 13 |- (mulGrp ` P ) = (mulGrp ` P )
43	42, 5	mgpbas 18495	. . . . . . . . . . . 12 |- B = ( Base ` (mulGrp ` P ) )
44		eqid 2622	. . . . . . . . . . . 12 |- (.g ` (mulGrp ` P ) ) = (.g ` (mulGrp ` P ) )
45	43, 44	mulg1 17548	. . . . . . . . . . 11 |- ( X e. B -> ( 1 (.g ` (mulGrp ` P ) ) X ) = X )
46	41, 45	syl 17	. . . . . . . . . 10 |- ( ph -> ( 1 (.g ` (mulGrp ` P ) ) X ) = X )
47	46	oveq2d 6666	. . . . . . . . 9 |- ( ph -> ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) = ( M .x. X ) )
48		fta1blem.5	. . . . . . . . . . 11 |- ( ph -> M =/= W )
49		fta1b.w	. . . . . . . . . . . . 13 |- W = ( 0g ` R )
50	49, 4, 3, 7, 10, 42, 44	coe1tmfv1 19644	. . . . . . . . . . . 12 |- ( ( R e. Ring /\ M e. K /\ 1 e. NN0 ) -> ( (coe1 ` ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = M )
51	39, 9, 37, 50	syl3anc 1326	. . . . . . . . . . 11 |- ( ph -> ( (coe1 ` ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = M )
52		fta1b.z	. . . . . . . . . . . . . . 15 |- .0. = ( 0g ` P )
53	3, 52, 49	coe1z 19633	. . . . . . . . . . . . . 14 |- ( R e. Ring -> (coe1 ` .0. ) = ( NN0 X. { W } ) )
54	39, 53	syl 17	. . . . . . . . . . . . 13 |- ( ph -> (coe1 ` .0. ) = ( NN0 X. { W } ) )
55	54	fveq1d 6193	. . . . . . . . . . . 12 |- ( ph -> ( (coe1 ` .0. ) ` 1 ) = ( ( NN0 X. { W } ) ` 1 ) )
56		fvex 6201	. . . . . . . . . . . . . . 15 |- ( 0g ` R ) e. _V
57	49, 56	eqeltri 2697	. . . . . . . . . . . . . 14 |- W e. _V
58	57	fvconst2 6469	. . . . . . . . . . . . 13 |- ( 1 e. NN0 -> ( ( NN0 X. { W } ) ` 1 ) = W )
59	36, 58	ax-mp 5	. . . . . . . . . . . 12 |- ( ( NN0 X. { W } ) ` 1 ) = W
60	55, 59	syl6eq 2672	. . . . . . . . . . 11 |- ( ph -> ( (coe1 ` .0. ) ` 1 ) = W )
61	48, 51, 60	3netr4d 2871	. . . . . . . . . 10 |- ( ph -> ( (coe1 ` ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) =/= ( (coe1 ` .0. ) ` 1 ) )
62		fveq2 6191	. . . . . . . . . . . 12 |- ( ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) = .0. -> (coe1 ` ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) = (coe1 ` .0. ) )
63	62	fveq1d 6193	. . . . . . . . . . 11 |- ( ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) = .0. -> ( (coe1 ` ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) = ( (coe1 ` .0. ) ` 1 ) )
64	63	necon3i 2826	. . . . . . . . . 10 |- ( ( (coe1 ` ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) ` 1 ) =/= ( (coe1 ` .0. ) ` 1 ) -> ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) =/= .0. )
65	61, 64	syl 17	. . . . . . . . 9 |- ( ph -> ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) =/= .0. )
66	47, 65	eqnetrrd 2862	. . . . . . . 8 |- ( ph -> ( M .x. X ) =/= .0. )
67		eldifsn 4317	. . . . . . . 8 |- ( ( M .x. X ) e. ( B \ { .0. } ) <-> ( ( M .x. X ) e. B /\ ( M .x. X ) =/= .0. ) )
68	25, 66, 67	sylanbrc 698	. . . . . . 7 |- ( ph -> ( M .x. X ) e. ( B \ { .0. } ) )
69		fta1blem.6	. . . . . . 7 |- ( ph -> ( ( M .x. X ) e. ( B \ { .0. } ) -> ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <_ ( D ` ( M .x. X ) ) ) )
70	68, 69	mpd 15	. . . . . 6 |- ( ph -> ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <_ ( D ` ( M .x. X ) ) )
71	47	fveq2d 6195	. . . . . . 7 |- ( ph -> ( D ` ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) = ( D ` ( M .x. X ) ) )
72		fta1b.d	. . . . . . . . 9 |- D = ( deg1 ` R )
73	72, 4, 3, 7, 10, 42, 44, 49	deg1tm 23878	. . . . . . . 8 |- ( ( R e. Ring /\ ( M e. K /\ M =/= W ) /\ 1 e. NN0 ) -> ( D ` ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) = 1 )
74	39, 9, 48, 37, 73	syl121anc 1331	. . . . . . 7 |- ( ph -> ( D ` ( M .x. ( 1 (.g ` (mulGrp ` P ) ) X ) ) ) = 1 )
75	71, 74	eqtr3d 2658	. . . . . 6 |- ( ph -> ( D ` ( M .x. X ) ) = 1 )
76	70, 75	breqtrd 4679	. . . . 5 |- ( ph -> ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <_ 1 )
77		hashbnd 13123	. . . . 5 |- ( ( ( `' ( O ` ( M .x. X ) ) " { W } ) e. _V /\ 1 e. NN0 /\ ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <_ 1 ) -> ( `' ( O ` ( M .x. X ) ) " { W } ) e. Fin )
78	35, 37, 76, 77	syl3anc 1326	. . . 4 |- ( ph -> ( `' ( O ` ( M .x. X ) ) " { W } ) e. Fin )
79	4, 49	ring0cl 18569	. . . . . . 7 |- ( R e. Ring -> W e. K )
80	39, 79	syl 17	. . . . . 6 |- ( ph -> W e. K )
81		eqid 2622	. . . . . . . . . . . . 13 |- (algSc ` P ) = (algSc ` P )
82	3, 81, 4, 5	ply1sclf 19655	. . . . . . . . . . . 12 |- ( R e. Ring -> (algSc ` P ) : K --> B )
83	39, 82	syl 17	. . . . . . . . . . 11 |- ( ph -> (algSc ` P ) : K --> B )
84	83, 9	ffvelrnd 6360	. . . . . . . . . 10 |- ( ph -> ( (algSc ` P ) ` M ) e. B )
85		eqid 2622	. . . . . . . . . . 11 |- ( .r ` P ) = ( .r ` P )
86		eqid 2622	. . . . . . . . . . 11 |- ( .r ` ( R ^s K ) ) = ( .r ` ( R ^s K ) )
87	5, 85, 86	rhmmul 18727	. . . . . . . . . 10 |- ( ( O e. ( P RingHom ( R ^s K ) ) /\ ( (algSc ` P ) ` M ) e. B /\ X e. B ) -> ( O ` ( ( (algSc ` P ) ` M ) ( .r ` P ) X ) ) = ( ( O ` ( (algSc ` P ) ` M ) ) ( .r ` ( R ^s K ) ) ( O ` X ) ) )
88	22, 84, 41, 87	syl3anc 1326	. . . . . . . . 9 |- ( ph -> ( O ` ( ( (algSc ` P ) ` M ) ( .r ` P ) X ) ) = ( ( O ` ( (algSc ` P ) ` M ) ) ( .r ` ( R ^s K ) ) ( O ` X ) ) )
89	3	ply1assa 19569	. . . . . . . . . . . 12 |- ( R e. CRing -> P e. AssAlg )
90	6, 89	syl 17	. . . . . . . . . . 11 |- ( ph -> P e. AssAlg )
91	3	ply1sca 19623	. . . . . . . . . . . . . . 15 |- ( R e. CRing -> R = (Scalar ` P ) )
92	6, 91	syl 17	. . . . . . . . . . . . . 14 |- ( ph -> R = (Scalar ` P ) )
93	92	fveq2d 6195	. . . . . . . . . . . . 13 |- ( ph -> ( Base ` R ) = ( Base ` (Scalar ` P ) ) )
94	4, 93	syl5eq 2668	. . . . . . . . . . . 12 |- ( ph -> K = ( Base ` (Scalar ` P ) ) )
95	9, 94	eleqtrd 2703	. . . . . . . . . . 11 |- ( ph -> M e. ( Base ` (Scalar ` P ) ) )
96		eqid 2622	. . . . . . . . . . . 12 |- (Scalar ` P ) = (Scalar ` P )
97		eqid 2622	. . . . . . . . . . . 12 |- ( Base ` (Scalar ` P ) ) = ( Base ` (Scalar ` P ) )
98	81, 96, 97, 5, 85, 10	asclmul1 19339	. . . . . . . . . . 11 |- ( ( P e. AssAlg /\ M e. ( Base ` (Scalar ` P ) ) /\ X e. B ) -> ( ( (algSc ` P ) ` M ) ( .r ` P ) X ) = ( M .x. X ) )
99	90, 95, 41, 98	syl3anc 1326	. . . . . . . . . 10 |- ( ph -> ( ( (algSc ` P ) ` M ) ( .r ` P ) X ) = ( M .x. X ) )
100	99	fveq2d 6195	. . . . . . . . 9 |- ( ph -> ( O ` ( ( (algSc ` P ) ` M ) ( .r ` P ) X ) ) = ( O ` ( M .x. X ) ) )
101	24, 84	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ph -> ( O ` ( (algSc ` P ) ` M ) ) e. ( Base ` ( R ^s K ) ) )
102	24, 41	ffvelrnd 6360	. . . . . . . . . . 11 |- ( ph -> ( O ` X ) e. ( Base ` ( R ^s K ) ) )
103	16, 17, 6, 20, 101, 102, 11, 86	pwsmulrval 16151	. . . . . . . . . 10 |- ( ph -> ( ( O ` ( (algSc ` P ) ` M ) ) ( .r ` ( R ^s K ) ) ( O ` X ) ) = ( ( O ` ( (algSc ` P ) ` M ) ) oF .X. ( O ` X ) ) )
104	2, 3, 4, 81	evl1sca 19698	. . . . . . . . . . . 12 |- ( ( R e. CRing /\ M e. K ) -> ( O ` ( (algSc ` P ) ` M ) ) = ( K X. { M } ) )
105	6, 9, 104	syl2anc 693	. . . . . . . . . . 11 |- ( ph -> ( O ` ( (algSc ` P ) ` M ) ) = ( K X. { M } ) )
106	2, 7, 4	evl1var 19700	. . . . . . . . . . . 12 |- ( R e. CRing -> ( O ` X ) = ( _I |` K ) )
107	6, 106	syl 17	. . . . . . . . . . 11 |- ( ph -> ( O ` X ) = ( _I |` K ) )
108	105, 107	oveq12d 6668	. . . . . . . . . 10 |- ( ph -> ( ( O ` ( (algSc ` P ) ` M ) ) oF .X. ( O ` X ) ) = ( ( K X. { M } ) oF .X. ( _I |` K ) ) )
109	103, 108	eqtrd 2656	. . . . . . . . 9 |- ( ph -> ( ( O ` ( (algSc ` P ) ` M ) ) ( .r ` ( R ^s K ) ) ( O ` X ) ) = ( ( K X. { M } ) oF .X. ( _I |` K ) ) )
110	88, 100, 109	3eqtr3d 2664	. . . . . . . 8 |- ( ph -> ( O ` ( M .x. X ) ) = ( ( K X. { M } ) oF .X. ( _I |` K ) ) )
111	110	fveq1d 6193	. . . . . . 7 |- ( ph -> ( ( O ` ( M .x. X ) ) ` W ) = ( ( ( K X. { M } ) oF .X. ( _I |` K ) ) ` W ) )
112		fnconstg 6093	. . . . . . . . . 10 |- ( M e. K -> ( K X. { M } ) Fn K )
113	9, 112	syl 17	. . . . . . . . 9 |- ( ph -> ( K X. { M } ) Fn K )
114		fnresi 6008	. . . . . . . . . 10 |- ( _I |` K ) Fn K
115	114	a1i 11	. . . . . . . . 9 |- ( ph -> ( _I |` K ) Fn K )
116		fnfvof 6911	. . . . . . . . 9 |- ( ( ( ( K X. { M } ) Fn K /\ ( _I |` K ) Fn K ) /\ ( K e. _V /\ W e. K ) ) -> ( ( ( K X. { M } ) oF .X. ( _I |` K ) ) ` W ) = ( ( ( K X. { M } ) ` W ) .X. ( ( _I |` K ) ` W ) ) )
117	113, 115, 20, 80, 116	syl22anc 1327	. . . . . . . 8 |- ( ph -> ( ( ( K X. { M } ) oF .X. ( _I |` K ) ) ` W ) = ( ( ( K X. { M } ) ` W ) .X. ( ( _I |` K ) ` W ) ) )
118		fvconst2g 6467	. . . . . . . . . . 11 |- ( ( M e. K /\ W e. K ) -> ( ( K X. { M } ) ` W ) = M )
119	9, 80, 118	syl2anc 693	. . . . . . . . . 10 |- ( ph -> ( ( K X. { M } ) ` W ) = M )
120		fvresi 6439	. . . . . . . . . . 11 |- ( W e. K -> ( ( _I |` K ) ` W ) = W )
121	80, 120	syl 17	. . . . . . . . . 10 |- ( ph -> ( ( _I |` K ) ` W ) = W )
122	119, 121	oveq12d 6668	. . . . . . . . 9 |- ( ph -> ( ( ( K X. { M } ) ` W ) .X. ( ( _I |` K ) ` W ) ) = ( M .X. W ) )
123	4, 11, 49	ringrz 18588	. . . . . . . . . 10 |- ( ( R e. Ring /\ M e. K ) -> ( M .X. W ) = W )
124	39, 9, 123	syl2anc 693	. . . . . . . . 9 |- ( ph -> ( M .X. W ) = W )
125	122, 124	eqtrd 2656	. . . . . . . 8 |- ( ph -> ( ( ( K X. { M } ) ` W ) .X. ( ( _I |` K ) ` W ) ) = W )
126	117, 125	eqtrd 2656	. . . . . . 7 |- ( ph -> ( ( ( K X. { M } ) oF .X. ( _I |` K ) ) ` W ) = W )
127	111, 126	eqtrd 2656	. . . . . 6 |- ( ph -> ( ( O ` ( M .x. X ) ) ` W ) = W )
128		fniniseg 6338	. . . . . . 7 |- ( ( O ` ( M .x. X ) ) Fn K -> ( W e. ( `' ( O ` ( M .x. X ) ) " { W } ) <-> ( W e. K /\ ( ( O ` ( M .x. X ) ) ` W ) = W ) ) )
129	28, 128	syl 17	. . . . . 6 |- ( ph -> ( W e. ( `' ( O ` ( M .x. X ) ) " { W } ) <-> ( W e. K /\ ( ( O ` ( M .x. X ) ) ` W ) = W ) ) )
130	80, 127, 129	mpbir2and 957	. . . . 5 |- ( ph -> W e. ( `' ( O ` ( M .x. X ) ) " { W } ) )
131	130	snssd 4340	. . . 4 |- ( ph -> { W } C_ ( `' ( O ` ( M .x. X ) ) " { W } ) )
132		hashsng 13159	. . . . . . 7 |- ( W e. K -> ( # ` { W } ) = 1 )
133	80, 132	syl 17	. . . . . 6 |- ( ph -> ( # ` { W } ) = 1 )
134		ssdomg 8001	. . . . . . . . . 10 |- ( ( `' ( O ` ( M .x. X ) ) " { W } ) e. _V -> ( { W } C_ ( `' ( O ` ( M .x. X ) ) " { W } ) -> { W } ~<_ ( `' ( O ` ( M .x. X ) ) " { W } ) ) )
135	34, 131, 134	mpsyl 68	. . . . . . . . 9 |- ( ph -> { W } ~<_ ( `' ( O ` ( M .x. X ) ) " { W } ) )
136		snfi 8038	. . . . . . . . . 10 |- { W } e. Fin
137		hashdom 13168	. . . . . . . . . 10 |- ( ( { W } e. Fin /\ ( `' ( O ` ( M .x. X ) ) " { W } ) e. _V ) -> ( ( # ` { W } ) <_ ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <-> { W } ~<_ ( `' ( O ` ( M .x. X ) ) " { W } ) ) )
138	136, 34, 137	mp2an 708	. . . . . . . . 9 |- ( ( # ` { W } ) <_ ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <-> { W } ~<_ ( `' ( O ` ( M .x. X ) ) " { W } ) )
139	135, 138	sylibr 224	. . . . . . . 8 |- ( ph -> ( # ` { W } ) <_ ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) )
140	133, 139	eqbrtrrd 4677	. . . . . . 7 |- ( ph -> 1 <_ ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) )
141		hashcl 13147	. . . . . . . . . 10 |- ( ( `' ( O ` ( M .x. X ) ) " { W } ) e. Fin -> ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) e. NN0 )
142	78, 141	syl 17	. . . . . . . . 9 |- ( ph -> ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) e. NN0 )
143	142	nn0red 11352	. . . . . . . 8 |- ( ph -> ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) e. RR )
144		1re 10039	. . . . . . . 8 |- 1 e. RR
145		letri3 10123	. . . . . . . 8 |- ( ( ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) e. RR /\ 1 e. RR ) -> ( ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) = 1 <-> ( ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <_ 1 /\ 1 <_ ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) ) ) )
146	143, 144, 145	sylancl 694	. . . . . . 7 |- ( ph -> ( ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) = 1 <-> ( ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <_ 1 /\ 1 <_ ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) ) ) )
147	76, 140, 146	mpbir2and 957	. . . . . 6 |- ( ph -> ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) = 1 )
148	133, 147	eqtr4d 2659	. . . . 5 |- ( ph -> ( # ` { W } ) = ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) )
149		hashen 13135	. . . . . 6 |- ( ( { W } e. Fin /\ ( `' ( O ` ( M .x. X ) ) " { W } ) e. Fin ) -> ( ( # ` { W } ) = ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <-> { W } ~~ ( `' ( O ` ( M .x. X ) ) " { W } ) ) )
150	136, 78, 149	sylancr 695	. . . . 5 |- ( ph -> ( ( # ` { W } ) = ( # ` ( `' ( O ` ( M .x. X ) ) " { W } ) ) <-> { W } ~~ ( `' ( O ` ( M .x. X ) ) " { W } ) ) )
151	148, 150	mpbid 222	. . . 4 |- ( ph -> { W } ~~ ( `' ( O ` ( M .x. X ) ) " { W } ) )
152		fisseneq 8171	. . . 4 |- ( ( ( `' ( O ` ( M .x. X ) ) " { W } ) e. Fin /\ { W } C_ ( `' ( O ` ( M .x. X ) ) " { W } ) /\ { W } ~~ ( `' ( O ` ( M .x. X ) ) " { W } ) ) -> { W } = ( `' ( O ` ( M .x. X ) ) " { W } ) )
153	78, 131, 151, 152	syl3anc 1326	. . 3 |- ( ph -> { W } = ( `' ( O ` ( M .x. X ) ) " { W } ) )
154	31, 153	eleqtrrd 2704	. 2 |- ( ph -> N e. { W } )
155		elsni 4194	. 2 |- ( N e. { W } -> N = W )
156	154, 155	syl 17	1 |- ( ph -> N = W )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   class class class wbr 4653   _I cid 5023   X. cxp 5112   `'ccnv 5113   |` cres 5116   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   ~~ cen 7952   ~<_ cdom 7953   Fincfn 7955   RRcr 9935   1c1 9937   <_ cle 10075   NN0cn0 11292   #chash 13117   Basecbs 15857   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   0gc0g 16100   ^_s cpws 16107  ._gcmg 17540  mulGrpcmgp 18489   Ringcrg 18547   CRingccrg 18548   RingHom crh 18712  AssAlgcasa 19309  algSccascl 19311  var₁cv1 19546  Poly₁cpl1 19547  coe₁cco1 19548  eval₁ce1 19679   deg₁ cdg1 23814

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  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

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-int 4476  df-iun 4522  df-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  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-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-xnn0 11364  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-evl 19507  df-psr1 19550  df-vr1 19551  df-ply1 19552  df-coe1 19553  df-evl1 19681  df-cnfld 19747  df-mdeg 23815  df-deg1 23816

This theorem is referenced by:  fta1b  23929