Mathbox for Jarvin Udandy |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > mdandysum2p2e4 | Structured version Visualization version GIF version |
Description: CONTRADICTION PROVED AT 1
+ 1 = 2 . Luckily Mario Carneiro did a
successful version of his own.
See Mario's Relevant Work: 1.3.14 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 which 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.) |
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)) |
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 ↔ ⊥))) |
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 39718 | . 2 ⊢ (𝜃 ↔ ⊥) |
7 | mdandysum2p2e4.e | . . 3 ⊢ (𝜏 ↔ jth) | |
8 | 7, 5 | aisbbisfaisf 39718 | . 2 ⊢ (𝜏 ↔ ⊥) |
9 | mdandysum2p2e4.f | . . 3 ⊢ (𝜂 ↔ jta) | |
10 | mdandysum2p2e4.2 | . . 3 ⊢ (jta ↔ ⊤) | |
11 | 9, 10 | aiffbbtat 39717 | . 2 ⊢ (𝜂 ↔ ⊤) |
12 | mdandysum2p2e4.g | . . 3 ⊢ (𝜁 ↔ jta) | |
13 | 12, 10 | aiffbbtat 39717 | . 2 ⊢ (𝜁 ↔ ⊤) |
14 | mdandysum2p2e4.h | . . 3 ⊢ (𝜎 ↔ jth) | |
15 | 14, 5 | aisbbisfaisf 39718 | . 2 ⊢ (𝜎 ↔ ⊥) |
16 | mdandysum2p2e4.i | . . 3 ⊢ (𝜌 ↔ jth) | |
17 | 16, 5 | aisbbisfaisf 39718 | . 2 ⊢ (𝜌 ↔ ⊥) |
18 | mdandysum2p2e4.j | . . 3 ⊢ (𝜇 ↔ jth) | |
19 | 18, 5 | aisbbisfaisf 39718 | . 2 ⊢ (𝜇 ↔ ⊥) |
20 | mdandysum2p2e4.k | . . 3 ⊢ (𝜆 ↔ jth) | |
21 | 20, 5 | aisbbisfaisf 39718 | . 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 39814 | 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 ↔ ⊥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ⊻ wxo 1456 ⊤wtru 1476 ⊥wfal 1480 |
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 196 df-or 384 df-an 385 df-xor 1457 df-tru 1478 df-fal 1481 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |