Theorem mdandysum2p2e4 44381

Description: CONTRADICTION PROVED AT 1 + 1 = 2 . Luckily Mario Carneiro did a successful version of his own.

See Mario's Relevant Work: Half adder and full adder in propositional calculus.

Given the right hypotheses we can prove a dandysum of 2+2=4. The qed step is the value '4' in Decimal BEING IMPLIED by the hypotheses.

Note: Values that when added would exceed a 4bit value are not supported.

Note: Digits begin from left (least) to right (greatest). E.g., 1000 would be '1', 0100 would be '2'. 0010 would be '4'.

How to perceive the hypotheses' bits in order: ( th <-> F. ), ( ta <-> F. ) Would be input value X's first bit, and input value Y's first bit.

( et <-> F. ), ( ze <-> F. ) would be input value X's second bit, and input value Y's second bit.

In mdandysum2p2e4, one might imagine what jth or jta could be then do the math with their truths. Also limited to the restriction jth, jta are having opposite truths equivalent to the stated truth constants. (Contributed by Jarvin Udandy, 6-Sep-2016.)

Hypotheses

Ref	Expression
mdandysum2p2e4.1	⊢ (jth ↔ ⊥)
mdandysum2p2e4.2	⊢ (jta ↔ ⊤)
mdandysum2p2e4.a	⊢ (𝜑 ↔ (𝜃 ∧ 𝜏))
mdandysum2p2e4.b	⊢ (𝜓 ↔ (𝜂 ∧ 𝜁))
mdandysum2p2e4.c	⊢ (𝜒 ↔ (𝜎 ∧ 𝜌))
mdandysum2p2e4.d	⊢ (𝜃 ↔ jth)
mdandysum2p2e4.e	⊢ (𝜏 ↔ jth)
mdandysum2p2e4.f	⊢ (𝜂 ↔ jta)
mdandysum2p2e4.g	⊢ (𝜁 ↔ jta)
mdandysum2p2e4.h	⊢ (𝜎 ↔ jth)
mdandysum2p2e4.i	⊢ (𝜌 ↔ jth)
mdandysum2p2e4.j	⊢ (𝜇 ↔ jth)
mdandysum2p2e4.k	⊢ (𝜆 ↔ jth)
mdandysum2p2e4.l	⊢ (𝜅 ↔ ((𝜃 ⊻ 𝜏) ⊻ (𝜃 ∧ 𝜏)))
mdandysum2p2e4.m	⊢ (jph ↔ ((𝜂 ⊻ 𝜁) ∨ 𝜑))
mdandysum2p2e4.n	⊢ (jps ↔ ((𝜎 ⊻ 𝜌) ∨ 𝜓))
mdandysum2p2e4.o	⊢ (jch ↔ ((𝜇 ⊻ 𝜆) ∨ 𝜒))

Assertion

Ref	Expression
mdandysum2p2e4	⊢ ((((((((((((((((𝜑 ↔ (𝜃 ∧ 𝜏)) ∧ (𝜓 ↔ (𝜂 ∧ 𝜁))) ∧ (𝜒 ↔ (𝜎 ∧ 𝜌))) ∧ (𝜃 ↔ ⊥)) ∧ (𝜏 ↔ ⊥)) ∧ (𝜂 ↔ ⊤)) ∧ (𝜁 ↔ ⊤)) ∧ (𝜎 ↔ ⊥)) ∧ (𝜌 ↔ ⊥)) ∧ (𝜇 ↔ ⊥)) ∧ (𝜆 ↔ ⊥)) ∧ (𝜅 ↔ ((𝜃 ⊻ 𝜏) ⊻ (𝜃 ∧ 𝜏)))) ∧ (jph ↔ ((𝜂 ⊻ 𝜁) ∨ 𝜑))) ∧ (jps ↔ ((𝜎 ⊻ 𝜌) ∨ 𝜓))) ∧ (jch ↔ ((𝜇 ⊻ 𝜆) ∨ 𝜒))) → ((((𝜅 ↔ ⊥) ∧ (jph ↔ ⊥)) ∧ (jps ↔ ⊤)) ∧ (jch ↔ ⊥)))

Proof of Theorem mdandysum2p2e4

Step	Hyp	Ref	Expression
1		mdandysum2p2e4.a	. 2 ⊢ (𝜑 ↔ (𝜃 ∧ 𝜏))
2		mdandysum2p2e4.b	. 2 ⊢ (𝜓 ↔ (𝜂 ∧ 𝜁))
3		mdandysum2p2e4.c	. 2 ⊢ (𝜒 ↔ (𝜎 ∧ 𝜌))
4		mdandysum2p2e4.d	. . 3 ⊢ (𝜃 ↔ jth)
5		mdandysum2p2e4.1	. . 3 ⊢ (jth ↔ ⊥)
6	4, 5	aisbbisfaisf 44284	. 2 ⊢ (𝜃 ↔ ⊥)
7		mdandysum2p2e4.e	. . 3 ⊢ (𝜏 ↔ jth)
8	7, 5	aisbbisfaisf 44284	. 2 ⊢ (𝜏 ↔ ⊥)
9		mdandysum2p2e4.f	. . 3 ⊢ (𝜂 ↔ jta)
10		mdandysum2p2e4.2	. . 3 ⊢ (jta ↔ ⊤)
11	9, 10	aiffbbtat 44283	. 2 ⊢ (𝜂 ↔ ⊤)
12		mdandysum2p2e4.g	. . 3 ⊢ (𝜁 ↔ jta)
13	12, 10	aiffbbtat 44283	. 2 ⊢ (𝜁 ↔ ⊤)
14		mdandysum2p2e4.h	. . 3 ⊢ (𝜎 ↔ jth)
15	14, 5	aisbbisfaisf 44284	. 2 ⊢ (𝜎 ↔ ⊥)
16		mdandysum2p2e4.i	. . 3 ⊢ (𝜌 ↔ jth)
17	16, 5	aisbbisfaisf 44284	. 2 ⊢ (𝜌 ↔ ⊥)
18		mdandysum2p2e4.j	. . 3 ⊢ (𝜇 ↔ jth)
19	18, 5	aisbbisfaisf 44284	. 2 ⊢ (𝜇 ↔ ⊥)
20		mdandysum2p2e4.k	. . 3 ⊢ (𝜆 ↔ jth)
21	20, 5	aisbbisfaisf 44284	. 2 ⊢ (𝜆 ↔ ⊥)
22		mdandysum2p2e4.l	. 2 ⊢ (𝜅 ↔ ((𝜃 ⊻ 𝜏) ⊻ (𝜃 ∧ 𝜏)))
23		mdandysum2p2e4.m	. 2 ⊢ (jph ↔ ((𝜂 ⊻ 𝜁) ∨ 𝜑))
24		mdandysum2p2e4.n	. 2 ⊢ (jps ↔ ((𝜎 ⊻ 𝜌) ∨ 𝜓))
25		mdandysum2p2e4.o	. 2 ⊢ (jch ↔ ((𝜇 ⊻ 𝜆) ∨ 𝜒))
26	1, 2, 3, 6, 8, 11, 13, 15, 17, 19, 21, 22, 23, 24, 25	dandysum2p2e4 44380	1 ⊢ ((((((((((((((((𝜑 ↔ (𝜃 ∧ 𝜏)) ∧ (𝜓 ↔ (𝜂 ∧ 𝜁))) ∧ (𝜒 ↔ (𝜎 ∧ 𝜌))) ∧ (𝜃 ↔ ⊥)) ∧ (𝜏 ↔ ⊥)) ∧ (𝜂 ↔ ⊤)) ∧ (𝜁 ↔ ⊤)) ∧ (𝜎 ↔ ⊥)) ∧ (𝜌 ↔ ⊥)) ∧ (𝜇 ↔ ⊥)) ∧ (𝜆 ↔ ⊥)) ∧ (𝜅 ↔ ((𝜃 ⊻ 𝜏) ⊻ (𝜃 ∧ 𝜏)))) ∧ (jph ↔ ((𝜂 ⊻ 𝜁) ∨ 𝜑))) ∧ (jps ↔ ((𝜎 ⊻ 𝜌) ∨ 𝜓))) ∧ (jch ↔ ((𝜇 ⊻ 𝜆) ∨ 𝜒))) → ((((𝜅 ↔ ⊥) ∧ (jph ↔ ⊥)) ∧ (jps ↔ ⊤)) ∧ (jch ↔ ⊥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843   ⊻ wxo 1503  ⊤wtru 1540  ⊥wfal 1551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-xor 1504  df-tru 1542  df-fal 1552

This theorem is referenced by: (None)