Theorem algrp1lem 13741

Description: Lemma for algrp1 13742.

Hypothesis

Ref	Expression
algrp1lem.1	|- Fun F

Assertion

Ref	Expression
algrp1lem	|- ((A e. S /\ B e. S /\ K e. NN0) -> (B(F o. (1st |` (S X. S)))((NN0 X. {A})` K)) = (F` B))

Proof of Theorem algrp1lem

Step	Hyp	Ref	Expression
1		opelxpi 4040	. . . . . . 7 |- ((B e. S /\ ((NN0 X. {A})` K) e. S) -> <.B, ((NN0 X. {A})` K)>. e. (S X. S))
2		fvconst2g 4820	. . . . . . . . . . 11 |- ((A e. S /\ K e. NN0) -> ((NN0 X. {A})` K) = A)
3	2	eleq1d 1963	. . . . . . . . . 10 |- ((A e. S /\ K e. NN0) -> (((NN0 X. {A})` K) e. S <-> A e. S))
4	3	biimprd 171	. . . . . . . . 9 |- ((A e. S /\ K e. NN0) -> (A e. S -> ((NN0 X. {A})` K) e. S))
5	4	adantrd 427	. . . . . . . 8 |- ((A e. S /\ K e. NN0) -> ((A e. S /\ K e. NN0) -> ((NN0 X. {A})` K) e. S))
6	5	pm2.43i 78	. . . . . . 7 |- ((A e. S /\ K e. NN0) -> ((NN0 X. {A})` K) e. S)
7	1, 6	sylan2 500	. . . . . 6 |- ((B e. S /\ (A e. S /\ K e. NN0)) -> <.B, ((NN0 X. {A})` K)>. e. (S X. S))
8	7	3impb 1063	. . . . 5 |- ((B e. S /\ A e. S /\ K e. NN0) -> <.B, ((NN0 X. {A})` K)>. e. (S X. S))
9	8	3com12 1071	. . . 4 |- ((A e. S /\ B e. S /\ K e. NN0) -> <.B, ((NN0 X. {A})` K)>. e. (S X. S))
10		algrp1lem.1	. . . . 5 |- Fun F
11		f1stres 5034	. . . . 5 |- (1st |` (S X. S)):(S X. S)-->S
12		fvco3 4739	. . . . 5 |- ((Fun F /\ (1st |` (S X. S)):(S X. S)-->S /\ <.B, ((NN0 X. {A})` K)>. e. (S X. S)) -> ((F o. (1st |` (S X. S)))` <.B, ((NN0 X. {A})` K)>.) = (F` ((1st |` (S X. S))` <.B, ((NN0 X. {A})` K)>.)))
13	10, 11, 12	mp3an12 1181	. . . 4 |- (<.B, ((NN0 X. {A})` K)>. e. (S X. S) -> ((F o. (1st |` (S X. S)))` <.B, ((NN0 X. {A})` K)>.) = (F` ((1st |` (S X. S))` <.B, ((NN0 X. {A})` K)>.)))
14	9, 13	syl 12	. . 3 |- ((A e. S /\ B e. S /\ K e. NN0) -> ((F o. (1st |` (S X. S)))` <.B, ((NN0 X. {A})` K)>.) = (F` ((1st |` (S X. S))` <.B, ((NN0 X. {A})` K)>.)))
15		fvres 4691	. . . . . 6 |- (<.B, ((NN0 X. {A})` K)>. e. (S X. S) -> ((1st |` (S X. S))` <.B, ((NN0 X. {A})` K)>.) = (1st` <.B, ((NN0 X. {A})` K)>.))
16	9, 15	syl 12	. . . . 5 |- ((A e. S /\ B e. S /\ K e. NN0) -> ((1st |` (S X. S))` <.B, ((NN0 X. {A})` K)>.) = (1st` <.B, ((NN0 X. {A})` K)>.))
17		op1stg 5028	. . . . . 6 |- (B e. S -> (1st` <.B, ((NN0 X. {A})` K)>.) = B)
18	17	3ad2ant2 898	. . . . 5 |- ((A e. S /\ B e. S /\ K e. NN0) -> (1st` <.B, ((NN0 X. {A})` K)>.) = B)
19	16, 18	eqtrd 1925	. . . 4 |- ((A e. S /\ B e. S /\ K e. NN0) -> ((1st |` (S X. S))` <.B, ((NN0 X. {A})` K)>.) = B)
20	19	fveq2d 4685	. . 3 |- ((A e. S /\ B e. S /\ K e. NN0) -> (F` ((1st |` (S X. S))` <.B, ((NN0 X. {A})` K)>.)) = (F` B))
21	14, 20	eqtrd 1925	. 2 |- ((A e. S /\ B e. S /\ K e. NN0) -> ((F o. (1st |` (S X. S)))` <.B, ((NN0 X. {A})` K)>.) = (F` B))
22		df-opr 4886	. 2 |- (B(F o. (1st |` (S X. S)))((NN0 X. {A})` K)) = ((F o. (1st |` (S X. S)))` <.B, ((NN0 X. {A})` K)>.)
23	21, 22	syl5eq 1940	1 |- ((A e. S /\ B e. S /\ K e. NN0) -> (B(F o. (1st |` (S X. S)))((NN0 X. {A})` K)) = (F` B))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  {csn 3044  <.cop 3046   X. cxp 3984   |` cres 3988   o. ccom 3990  Fun wfun 3992  -->wf 3994  ` cfv 3998  (class class class)co 4884  1stc1st 5018  NN0cn0 6450

This theorem is referenced by:  algrp1 13742

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014  df-opr 4886  df-1st 5020

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