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Theorem lautco 33127
 Description: The composition of two lattice automorphisms is a lattice automorphism. (Contributed by NM, 19-Apr-2013.)
Hypothesis
Ref Expression
lautco.i
Assertion
Ref Expression
lautco

Proof of Theorem lautco
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2404 . . . . 5
2 lautco.i . . . . 5
31, 2laut1o 33115 . . . 4
51, 2laut1o 33115 . . . 4
7 f1oco 5823 . . 3
84, 6, 7syl2anc 661 . 2
9 simpl1 1002 . . . . 5
10 simpl2 1003 . . . . 5
11 simpl3 1004 . . . . . 6
12 simprl 758 . . . . . 6
131, 2lautcl 33117 . . . . . 6
149, 11, 12, 13syl21anc 1231 . . . . 5
15 simprr 760 . . . . . 6
161, 2lautcl 33117 . . . . . 6
179, 11, 15, 16syl21anc 1231 . . . . 5
18 eqid 2404 . . . . . 6
191, 18, 2lautle 33114 . . . . 5
209, 10, 14, 17, 19syl22anc 1233 . . . 4
211, 18, 2lautle 33114 . . . . 5
22213adantl2 1156 . . . 4
23 f1of 5801 . . . . . . 7
246, 23syl 17 . . . . . 6
25 simpl 457 . . . . . 6
26 fvco3 5928 . . . . . 6
2724, 25, 26syl2an 477 . . . . 5
28 simpr 461 . . . . . 6
29 fvco3 5928 . . . . . 6
3024, 28, 29syl2an 477 . . . . 5
3127, 30breq12d 4410 . . . 4
3220, 22, 313bitr4d 287 . . 3
3332ralrimivva 2827 . 2
341, 18, 2islaut 33113 . . 3