Theorem sralem 19177

Description: Lemma for srabase 19178 and similar theorems. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)

Hypotheses

Ref	Expression
srapart.a	|- ( ph -> A = ( (subringAlg ` W ) ` S ) )
srapart.s	|- ( ph -> S C_ ( Base ` W ) )
sralem.1	|- E = Slot N
sralem.2	|- N e. NN
sralem.3	|- ( N < 5 \/ 8 < N )

Assertion

Ref	Expression
sralem	|- ( ph -> ( E ` W ) = ( E ` A ) )

Proof of Theorem sralem

Step	Hyp	Ref	Expression
1		srapart.a	. . . . . 6 |- ( ph -> A = ( (subringAlg ` W ) ` S ) )
2	1	adantl 482	. . . . 5 |- ( ( W e. _V /\ ph ) -> A = ( (subringAlg ` W ) ` S ) )
3		srapart.s	. . . . . 6 |- ( ph -> S C_ ( Base ` W ) )
4		sraval 19176	. . . . . 6 |- ( ( W e. _V /\ S C_ ( Base ` W ) ) -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
5	3, 4	sylan2 491	. . . . 5 |- ( ( W e. _V /\ ph ) -> ( (subringAlg ` W ) ` S ) = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
6	2, 5	eqtrd 2656	. . . 4 |- ( ( W e. _V /\ ph ) -> A = ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
7	6	fveq2d 6195	. . 3 |- ( ( W e. _V /\ ph ) -> ( E ` A ) = ( E ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) ) )
8		sralem.1	. . . . . 6 |- E = Slot N
9		sralem.2	. . . . . 6 |- N e. NN
10	8, 9	ndxid 15883	. . . . 5 |- E = Slot ( E ` ndx )
11		sralem.3	. . . . . . 7 |- ( N < 5 \/ 8 < N )
12	9	nnrei 11029	. . . . . . . . . 10 |- N e. RR
13		5re 11099	. . . . . . . . . 10 |- 5 e. RR
14	12, 13	ltnei 10161	. . . . . . . . 9 |- ( N < 5 -> 5 =/= N )
15	14	necomd 2849	. . . . . . . 8 |- ( N < 5 -> N =/= 5 )
16		5lt8 11217	. . . . . . . . . 10 |- 5 < 8
17		8re 11105	. . . . . . . . . . 11 |- 8 e. RR
18	13, 17, 12	lttri 10163	. . . . . . . . . 10 |- ( ( 5 < 8 /\ 8 < N ) -> 5 < N )
19	16, 18	mpan 706	. . . . . . . . 9 |- ( 8 < N -> 5 < N )
20	13, 12	ltnei 10161	. . . . . . . . 9 |- ( 5 < N -> N =/= 5 )
21	19, 20	syl 17	. . . . . . . 8 |- ( 8 < N -> N =/= 5 )
22	15, 21	jaoi 394	. . . . . . 7 |- ( ( N < 5 \/ 8 < N ) -> N =/= 5 )
23	11, 22	ax-mp 5	. . . . . 6 |- N =/= 5
24	8, 9	ndxarg 15882	. . . . . . 7 |- ( E ` ndx ) = N
25		scandx 16013	. . . . . . 7 |- (Scalar ` ndx ) = 5
26	24, 25	neeq12i 2860	. . . . . 6 |- ( ( E ` ndx ) =/= (Scalar ` ndx ) <-> N =/= 5 )
27	23, 26	mpbir 221	. . . . 5 |- ( E ` ndx ) =/= (Scalar ` ndx )
28	10, 27	setsnid 15915	. . . 4 |- ( E ` W ) = ( E ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) )
29		5lt6 11204	. . . . . . . . . . 11 |- 5 < 6
30		6re 11101	. . . . . . . . . . . 12 |- 6 e. RR
31	12, 13, 30	lttri 10163	. . . . . . . . . . 11 |- ( ( N < 5 /\ 5 < 6 ) -> N < 6 )
32	29, 31	mpan2 707	. . . . . . . . . 10 |- ( N < 5 -> N < 6 )
33	12, 30	ltnei 10161	. . . . . . . . . 10 |- ( N < 6 -> 6 =/= N )
34	32, 33	syl 17	. . . . . . . . 9 |- ( N < 5 -> 6 =/= N )
35	34	necomd 2849	. . . . . . . 8 |- ( N < 5 -> N =/= 6 )
36		6lt8 11216	. . . . . . . . . 10 |- 6 < 8
37	30, 17, 12	lttri 10163	. . . . . . . . . 10 |- ( ( 6 < 8 /\ 8 < N ) -> 6 < N )
38	36, 37	mpan 706	. . . . . . . . 9 |- ( 8 < N -> 6 < N )
39	30, 12	ltnei 10161	. . . . . . . . 9 |- ( 6 < N -> N =/= 6 )
40	38, 39	syl 17	. . . . . . . 8 |- ( 8 < N -> N =/= 6 )
41	35, 40	jaoi 394	. . . . . . 7 |- ( ( N < 5 \/ 8 < N ) -> N =/= 6 )
42	11, 41	ax-mp 5	. . . . . 6 |- N =/= 6
43		vscandx 16015	. . . . . . 7 |- ( .s ` ndx ) = 6
44	24, 43	neeq12i 2860	. . . . . 6 |- ( ( E ` ndx ) =/= ( .s ` ndx ) <-> N =/= 6 )
45	42, 44	mpbir 221	. . . . 5 |- ( E ` ndx ) =/= ( .s ` ndx )
46	10, 45	setsnid 15915	. . . 4 |- ( E ` ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) ) = ( E ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) )
47	12, 13, 17	lttri 10163	. . . . . . . . . . 11 |- ( ( N < 5 /\ 5 < 8 ) -> N < 8 )
48	16, 47	mpan2 707	. . . . . . . . . 10 |- ( N < 5 -> N < 8 )
49	12, 17	ltnei 10161	. . . . . . . . . 10 |- ( N < 8 -> 8 =/= N )
50	48, 49	syl 17	. . . . . . . . 9 |- ( N < 5 -> 8 =/= N )
51	50	necomd 2849	. . . . . . . 8 |- ( N < 5 -> N =/= 8 )
52	17, 12	ltnei 10161	. . . . . . . 8 |- ( 8 < N -> N =/= 8 )
53	51, 52	jaoi 394	. . . . . . 7 |- ( ( N < 5 \/ 8 < N ) -> N =/= 8 )
54	11, 53	ax-mp 5	. . . . . 6 |- N =/= 8
55		ipndx 16022	. . . . . . 7 |- ( .i ` ndx ) = 8
56	24, 55	neeq12i 2860	. . . . . 6 |- ( ( E ` ndx ) =/= ( .i ` ndx ) <-> N =/= 8 )
57	54, 56	mpbir 221	. . . . 5 |- ( E ` ndx ) =/= ( .i ` ndx )
58	10, 57	setsnid 15915	. . . 4 |- ( E ` ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) ) = ( E ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
59	28, 46, 58	3eqtri 2648	. . 3 |- ( E ` W ) = ( E ` ( ( ( W sSet <. (Scalar ` ndx ) , ( W ↾s S ) >. ) sSet <. ( .s ` ndx ) , ( .r ` W ) >. ) sSet <. ( .i ` ndx ) , ( .r ` W ) >. ) )
60	7, 59	syl6reqr 2675	. 2 |- ( ( W e. _V /\ ph ) -> ( E ` W ) = ( E ` A ) )
61	8	str0 15911	. . 3 |- (/) = ( E ` (/) )
62		fvprc 6185	. . . 4 |- ( -. W e. _V -> ( E ` W ) = (/) )
63	62	adantr 481	. . 3 |- ( ( -. W e. _V /\ ph ) -> ( E ` W ) = (/) )
64		fvprc 6185	. . . . . . 7 |- ( -. W e. _V -> (subringAlg ` W ) = (/) )
65	64	fveq1d 6193	. . . . . 6 |- ( -. W e. _V -> ( (subringAlg ` W ) ` S ) = ( (/) ` S ) )
66		0fv 6227	. . . . . 6 |- ( (/) ` S ) = (/)
67	65, 66	syl6eq 2672	. . . . 5 |- ( -. W e. _V -> ( (subringAlg ` W ) ` S ) = (/) )
68	1, 67	sylan9eqr 2678	. . . 4 |- ( ( -. W e. _V /\ ph ) -> A = (/) )
69	68	fveq2d 6195	. . 3 |- ( ( -. W e. _V /\ ph ) -> ( E ` A ) = ( E ` (/) ) )
70	61, 63, 69	3eqtr4a 2682	. 2 |- ( ( -. W e. _V /\ ph ) -> ( E ` W ) = ( E ` A ) )
71	60, 70	pm2.61ian 831	1 |- ( ph -> ( E ` W ) = ( E ` A ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   _Vcvv 3200   C_ wss 3574   (/)c0 3915   <.cop 4183   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   < clt 10074   NNcn 11020   5c5 11073   6c6 11074   8c8 11076   ndxcnx 15854   sSet csts 15855  Slot cslot 15856   Basecbs 15857   ↾_s cress 15858   .rcmulr 15942  Scalarcsca 15944   .scvsca 15945   .icip 15946  subringAlg csra 19168

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-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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  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-ndx 15860  df-slot 15861  df-sets 15864  df-sca 15957  df-vsca 15958  df-ip 15959  df-sra 19172

This theorem is referenced by:  srabase  19178  sraaddg  19179  sramulr  19180  sratset  19184  srads  19186  cchhllem  25767