Theorem isassintop 48171

Description: The predicate "is an associative (closed internal binary) operations for a set". (Contributed by FL, 2-Nov-2009.) (Revised by AV, 20-Jan-2020.)

Assertion

Ref	Expression
isassintop	⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))

Distinct variable groups:   𝑥,𝑀,𝑦,𝑧   𝑥, ⚬ ,𝑦,𝑧

Allowed substitution hints:   𝑉(𝑥,𝑦,𝑧)

Proof of Theorem isassintop

Dummy variable 𝑜 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		assintopmap 48167	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
2	1	eleq2d 2814	. . . 4 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
3		breq1 5105	. . . . 5 ⊢ (𝑜 = ⚬ → (𝑜 assLaw 𝑀 ↔ ⚬ assLaw 𝑀))
4	3	elrab 3656	. . . 4 ⊢ ( ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀))
5	2, 4	bitrdi 287	. . 3 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀)))
6		elmapi 8799	. . . . . 6 ⊢ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
7	6	ad2antrl 728	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
8		isasslaw 48153	. . . . . . . 8 ⊢ (( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ 𝑀 ∈ 𝑉) → ( ⚬ assLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
9	8	biimpd 229	. . . . . . 7 ⊢ (( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ 𝑀 ∈ 𝑉) → ( ⚬ assLaw 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
10	9	impancom 451	. . . . . 6 ⊢ (( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀) → (𝑀 ∈ 𝑉 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
11	10	impcom 407	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀)) → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))
12	7, 11	jca 511	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀)) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
13	12	ex 412	. . 3 ⊢ (𝑀 ∈ 𝑉 → (( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))
14	5, 13	sylbid 240	. 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))
15		isclintop 48168	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
16	15	biimprcd 250	. . . . . 6 ⊢ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 → (𝑀 ∈ 𝑉 → ⚬ ∈ ( clIntOp ‘𝑀)))
17	16	adantr 480	. . . . 5 ⊢ (( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))) → (𝑀 ∈ 𝑉 → ⚬ ∈ ( clIntOp ‘𝑀)))
18	17	impcom 407	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) → ⚬ ∈ ( clIntOp ‘𝑀))
19		sqxpexg 7711	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
20		fex 7182	. . . . . . . . 9 ⊢ (( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ (𝑀 × 𝑀) ∈ V) → ⚬ ∈ V)
21	19, 20	sylan2 593	. . . . . . . 8 ⊢ (( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ 𝑀 ∈ 𝑉) → ⚬ ∈ V)
22	21	ancoms 458	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ ⚬ :(𝑀 × 𝑀)⟶𝑀) → ⚬ ∈ V)
23		simpl 482	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ ⚬ :(𝑀 × 𝑀)⟶𝑀) → 𝑀 ∈ 𝑉)
24		isasslaw 48153	. . . . . . . 8 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ 𝑉) → ( ⚬ assLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
25	24	bicomd 223	. . . . . . 7 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ 𝑉) → (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ⚬ assLaw 𝑀))
26	22, 23, 25	syl2anc 584	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ ⚬ :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ⚬ assLaw 𝑀))
27	26	biimpd 229	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ ⚬ :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) → ⚬ assLaw 𝑀))
28	27	impr 454	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) → ⚬ assLaw 𝑀)
29		assintopval 48166	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
30	29	adantr 480	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
31	30	eleq2d 2814	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ⚬ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀}))
32	3	elrab 3656	. . . . 5 ⊢ ( ⚬ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀))
33	31, 32	bitrdi 287	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀)))
34	18, 28, 33	mpbir2and 713	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) → ⚬ ∈ ( assIntOp ‘𝑀))
35	34	ex 412	. 2 ⊢ (𝑀 ∈ 𝑉 → (( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))) → ⚬ ∈ ( assIntOp ‘𝑀)))
36	14, 35	impbid 212	1 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3402  Vcvv 3444   class class class wbr 5102   × cxp 5629  ⟶wf 6495  ‘cfv 6499  (class class class)co 7369   ↑_m cmap 8776   assLaw casslaw 48145   clIntOp cclintop 48158   assIntOp cassintop 48159

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-1st 7947  df-2nd 7948  df-map 8778  df-asslaw 48149  df-intop 48160  df-clintop 48161  df-assintop 48162

This theorem is referenced by: (None)