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Theorem ndmovass 7621

Description: Any operation is associative outside its domain, if the domain doesn't contain the empty set. (Contributed by NM, 24-Aug-1995.)

**Hypotheses**
Ref	Expression
ndmov.1	⊢ dom 𝐹 = (𝑆 × 𝑆)
ndmov.5	⊢ ¬ ∅ ∈ 𝑆

**Assertion**
Ref	Expression
ndmovass	⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)))

**Proof of Theorem ndmovass**
Step	Hyp	Ref	Expression
1		ndmov.1	. . . . . 6 ⊢ dom 𝐹 = (𝑆 × 𝑆)
2		ndmov.5	. . . . . 6 ⊢ ¬ ∅ ∈ 𝑆
3	1, 2	ndmovrcl 7619	. . . . 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 1089	. . . 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 7617	. . 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 7619	. . . . 5 ⊢ ((𝐵𝐹𝐶) ∈ 𝑆 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))
10	9	anim2i 617	. . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵𝐹𝐶) ∈ 𝑆) → (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)))
11		3anass 1095	. . . 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 7617	. . 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 2780	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 1087 = wceq 1540 ∈ wcel 2108 ∅c0 4333 × cxp 5683 dom cdm 5685 (class class class)co 7431

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 df-iota 6514 df-fv 6569 df-ov 7434

This theorem is referenced by: addasspi 10935 mulasspi 10937 addassnq 10998 mulassnq 10999 genpass 11049 addasssr 11128 mulasssr 11130

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