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Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-1mintru1 | Structured version Visualization version GIF version |
Description: Using the recursion
formula:
"(n+1)-mintru-(m+1)" ↔ if-(𝜑, "n-mintru-m" , "n-mintru-(m+1)" ) for "1-mintru-1" (meaning "at least 1 out of 1 input is true") by plugging in n = 0, m = 0, and simplifying. The expressions "0-mintru-0" and "0-mintru-1" are base cases of the recursion, meaning "in a sequence of zero inputs, at least 0 / 1 input is true", respectively equvalent to ⊤ / ⊥. Negating an "n-mintru1" operation means: All n inputs 𝜑.. 𝜃 are false. This is also conveniently expressed as ¬ (𝜑 ∨.. ∨ 𝜃). Applying this idea here (n = 1) yields the obvious result that in an input sequence of size 1 only then all will be false, if its single input is. (Contributed by Wolf Lammen, 10-May-2024.) |
Ref | Expression |
---|---|
wl-1mintru1 | ⊢ (if-(𝜒, ⊤, ⊥) ↔ 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tbtru 1547 | . . . 4 ⊢ (𝜒 ↔ (𝜒 ↔ ⊤)) | |
2 | 1 | biimpi 219 | . . 3 ⊢ (𝜒 → (𝜒 ↔ ⊤)) |
3 | nbfal 1554 | . . . 4 ⊢ (¬ 𝜒 ↔ (𝜒 ↔ ⊥)) | |
4 | 3 | biimpi 219 | . . 3 ⊢ (¬ 𝜒 → (𝜒 ↔ ⊥)) |
5 | 2, 4 | casesifp 1075 | . 2 ⊢ (𝜒 ↔ if-(𝜒, ⊤, ⊥)) |
6 | 5 | bicomi 227 | 1 ⊢ (if-(𝜒, ⊤, ⊥) ↔ 𝜒) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 209 if-wif 1059 ⊤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 210 df-an 401 df-or 846 df-ifp 1060 df-tru 1542 df-fal 1552 |
This theorem is referenced by: (None) |
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