Theorem anval 148

Description: Value of the conjunction. (Contributed by Mario Carneiro, 9-Oct-2014.)

Hypotheses

Ref	Expression
imval.1	⊢ A:∗
imval.2	⊢ B:∗

Assertion

Ref	Expression
anval	⊢ ⊤⊧[[A ∧ B] = [λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))B] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]

Distinct variable groups:   f,A   f,B

Proof of Theorem anval

Dummy variables p q are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wan 136	. . 3 ⊢ ∧ :(∗ → (∗ → ∗))
2		imval.1	. . 3 ⊢ A:∗
3		imval.2	. . 3 ⊢ B:∗
4	1, 2, 3	wov 72	. 2 ⊢ [A ∧ B]:∗
5		df-an 128	. . 3 ⊢ ⊤⊧[ ∧ = λp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
6	1, 2, 3, 5	oveq 102	. 2 ⊢ ⊤⊧[[A ∧ B] = [Aλp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]B]]
7		wv 64	. . . . . 6 ⊢ f:(∗ → (∗ → ∗)):(∗ → (∗ → ∗))
8		wv 64	. . . . . 6 ⊢ p:∗:∗
9		wv 64	. . . . . 6 ⊢ q:∗:∗
10	7, 8, 9	wov 72	. . . . 5 ⊢ [p:∗f:(∗ → (∗ → ∗))q:∗]:∗
11	10	wl 66	. . . 4 ⊢ λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗]:((∗ → (∗ → ∗)) → ∗)
12		wtru 43	. . . . . 6 ⊢ ⊤:∗
13	7, 12, 12	wov 72	. . . . 5 ⊢ [⊤f:(∗ → (∗ → ∗))⊤]:∗
14	13	wl 66	. . . 4 ⊢ λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]:((∗ → (∗ → ∗)) → ∗)
15	11, 14	weqi 76	. . 3 ⊢ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]:∗
16		weq 41	. . . 4 ⊢ = :(((∗ → (∗ → ∗)) → ∗) → (((∗ → (∗ → ∗)) → ∗) → ∗))
17	8, 2	weqi 76	. . . . . . 7 ⊢ [p:∗ = A]:∗
18	17	id 25	. . . . . 6 ⊢ [p:∗ = A]⊧[p:∗ = A]
19	7, 8, 9, 18	oveq1 99	. . . . 5 ⊢ [p:∗ = A]⊧[[p:∗f:(∗ → (∗ → ∗))q:∗] = [Af:(∗ → (∗ → ∗))q:∗]]
20	10, 19	leq 91	. . . 4 ⊢ [p:∗ = A]⊧[λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))q:∗]]
21	16, 11, 14, 20	oveq1 99	. . 3 ⊢ [p:∗ = A]⊧[[λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]] = [λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
22	7, 2, 9	wov 72	. . . . 5 ⊢ [Af:(∗ → (∗ → ∗))q:∗]:∗
23	22	wl 66	. . . 4 ⊢ λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))q:∗]:((∗ → (∗ → ∗)) → ∗)
24	9, 3	weqi 76	. . . . . . 7 ⊢ [q:∗ = B]:∗
25	24	id 25	. . . . . 6 ⊢ [q:∗ = B]⊧[q:∗ = B]
26	7, 2, 9, 25	oveq2 101	. . . . 5 ⊢ [q:∗ = B]⊧[[Af:(∗ → (∗ → ∗))q:∗] = [Af:(∗ → (∗ → ∗))B]]
27	22, 26	leq 91	. . . 4 ⊢ [q:∗ = B]⊧[λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))B]]
28	16, 23, 14, 27	oveq1 99	. . 3 ⊢ [q:∗ = B]⊧[[λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]] = [λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))B] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
29	15, 2, 3, 21, 28	ovl 117	. 2 ⊢ ⊤⊧[[Aλp:∗ λq:∗ [λf:(∗ → (∗ → ∗)) [p:∗f:(∗ → (∗ → ∗))q:∗] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]B] = [λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))B] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]
30	4, 6, 29	eqtri 95	1 ⊢ ⊤⊧[[A ∧ B] = [λf:(∗ → (∗ → ∗)) [Af:(∗ → (∗ → ∗))B] = λf:(∗ → (∗ → ∗)) [⊤f:(∗ → (∗ → ∗))⊤]]]

Syntax hints:  tv 1   → ht 2  ∗hb 3  λkl 6   = ke 7  ⊤kt 8  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12   ∧ tan 119

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113

This theorem depends on definitions:  df-ov 73  df-an 128

This theorem is referenced by:  dfan2  154