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Mirrors > Home > MPE Home > Th. List > ndmovass | Structured version Visualization version GIF version |
Description: Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.) |
Ref | Expression |
---|---|
ndmov.1 | ⊢ dom 𝐹 = (𝑆 × 𝑆) |
ndmov.5 | ⊢ ¬ ∅ ∈ 𝑆 |
Ref | Expression |
---|---|
ndmovass | ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ndmov.1 | . . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆) | |
2 | ndmov.5 | . . . . . 6 ⊢ ¬ ∅ ∈ 𝑆 | |
3 | 1, 2 | ndmovrcl 7618 | . . . . 5 ⊢ ((𝐴𝐹𝐵) ∈ 𝑆 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) |
4 | 3 | anim1i 615 | . . . 4 ⊢ (((𝐴𝐹𝐵) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆)) |
5 | df-3an 1088 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) ∧ 𝐶 ∈ 𝑆)) | |
6 | 4, 5 | sylibr 234 | . . 3 ⊢ (((𝐴𝐹𝐵) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) |
7 | 1 | ndmov 7616 | . . 3 ⊢ (¬ ((𝐴𝐹𝐵) ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = ∅) |
8 | 6, 7 | nsyl5 159 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = ∅) |
9 | 1, 2 | ndmovrcl 7618 | . . . . 5 ⊢ ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) |
10 | 9 | anim2i 617 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))) |
11 | 3anass 1094 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ↔ (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))) | |
12 | 10, 11 | sylibr 234 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) |
13 | 1 | ndmov 7616 | . . 3 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴𝐹(𝐵𝐹𝐶)) = ∅) |
14 | 12, 13 | nsyl5 159 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → (𝐴𝐹(𝐵𝐹𝐶)) = ∅) |
15 | 8, 14 | eqtr4d 2777 | 1 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ∅c0 4338 × cxp 5686 dom cdm 5688 (class class class)co 7430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-xp 5694 df-dm 5698 df-iota 6515 df-fv 6570 df-ov 7433 |
This theorem is referenced by: addasspi 10932 mulasspi 10934 addassnq 10995 mulassnq 10996 genpass 11046 addasssr 11125 mulasssr 11127 |
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