Theorem resabs1 2592

Description: Absorption law for restriction. Exercise 17 of [TakeutiZaring] p. 25.

Assertion

Ref	Expression
resabs1	|- (B (_ C -> ((A |` C) |` B) = (A |` B))

Proof of Theorem resabs1

Step	Hyp	Ref	Expression
1		sseqin2 1656	. . . . . 6 |- (B (_ C <-> (C i^i B) = B)
2		xpeq1 2440	. . . . . 6 |- ((C i^i B) = B -> ((C i^i B) X. V) = (B X. V))
3	1, 2	sylbi 174	. . . . 5 |- (B (_ C -> ((C i^i B) X. V) = (B X. V))
4		inxp 2496	. . . . . 6 |- ((C X. V) i^i (B X. V)) = ((C i^i B) X. (V i^i V))
5		inidm 1649	. . . . . . 7 |- (V i^i V) = V
6		xpeq2 2441	. . . . . . 7 |- ((V i^i V) = V -> ((C i^i B) X. (V i^i V)) = ((C i^i B) X. V))
7	5, 6	ax-mp 6	. . . . . 6 |- ((C i^i B) X. (V i^i V)) = ((C i^i B) X. V)
8	4, 7	eqtr2 1120	. . . . 5 |- ((C i^i B) X. V) = ((C X. V) i^i (B X. V))
9	3, 8	syl5reqr 1139	. . . 4 |- (B (_ C -> (B X. V) = ((C X. V) i^i (B X. V)))
10	9	ineq2d 1645	. . 3 |- (B (_ C -> (A i^i (B X. V)) = (A i^i ((C X. V) i^i (B X. V))))
11		inass 1650	. . 3 |- ((A i^i (C X. V)) i^i (B X. V)) = (A i^i ((C X. V) i^i (B X. V)))
12	10, 11	syl6reqr 1143	. 2 |- (B (_ C -> ((A i^i (C X. V)) i^i (B X. V)) = (A i^i (B X. V)))
13		df-res 2430	. . 3 |- ((A |` C) |` B) = ((A |` C) i^i (B X. V))
14		df-res 2430	. . . 4 |- (A |` C) = (A i^i (C X. V))
15	14	ineq1i 1641	. . 3 |- ((A |` C) i^i (B X. V)) = ((A i^i (C X. V)) i^i (B X. V))
16	13, 15	eqtr 1119	. 2 |- ((A |` C) |` B) = ((A i^i (C X. V)) i^i (B X. V))
17		df-res 2430	. 2 |- (A |` B) = (A i^i (B X. V))
18	12, 16, 17	3eqtr4g 1147	1 |- (B (_ C -> ((A |` C) |` B) = (A |` B))

Syntax hints:   -> wi 2   = wceq 1091  Vcvv 1348   i^i cin 1486   (_ wss 1487   X. cxp 2408   |` cres 2412

This theorem is referenced by:  fun2ssres 2699  fvres 2840  tfrlem5 2953

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-xp 2424  df-rel 2425  df-res 2430

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