Theorem isassintop 45292

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 45288	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
2	1	eleq2d 2824	. . . 4 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
3		breq1 5073	. . . . 5 ⊢ (𝑜 = ⚬ → (𝑜 assLaw 𝑀 ↔ ⚬ assLaw 𝑀))
4	3	elrab 3617	. . . 4 ⊢ ( ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀))
5	2, 4	bitrdi 286	. . 3 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀)))
6		elmapi 8595	. . . . . 6 ⊢ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
7	6	ad2antrl 724	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
8		isasslaw 45274	. . . . . . . 8 ⊢ (( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ 𝑀 ∈ 𝑉) → ( ⚬ assLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
9	8	biimpd 228	. . . . . . 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 239	. 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))
15		isclintop 45289	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
16	15	biimprcd 249	. . . . . 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 7583	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
20		fex 7084	. . . . . . . . 9 ⊢ (( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ (𝑀 × 𝑀) ∈ V) → ⚬ ∈ V)
21	19, 20	sylan2 592	. . . . . . . 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 45274	. . . . . . . 8 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ 𝑉) → ( ⚬ assLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
25	24	bicomd 222	. . . . . . 7 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ 𝑉) → (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ⚬ assLaw 𝑀))
26	22, 23, 25	syl2anc 583	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ ⚬ :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ⚬ assLaw 𝑀))
27	26	biimpd 228	. . . . 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 45287	. . . . . . 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 2824	. . . . 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 3617	. . . . 5 ⊢ ( ⚬ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀))
33	31, 32	bitrdi 286	. . . 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 709	. . 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 211	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 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422   class class class wbr 5070   × cxp 5578  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573   assLaw casslaw 45266   clIntOp cclintop 45279   assIntOp cassintop 45280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575  df-asslaw 45270  df-intop 45281  df-clintop 45282  df-assintop 45283

This theorem is referenced by: (None)