Theorem isassintop 49128

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 49124	. . . . 5 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀})
2	1	eleq2d 2846	. . . 4 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀}))
3		breq1 5106	. . . . 5 ⊢ (𝑜 = ⚬ → (𝑜 assLaw 𝑀 ↔ ⚬ assLaw 𝑀))
4	3	elrab 3645	. . . 4 ⊢ ( ⚬ ∈ {𝑜 ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀))
5	2, 4	bitrdi 290	. . 3 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀)))
6		elmapi 8851	. . . . . 6 ⊢ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
7	6	ad2antrl 741	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀)) → ⚬ :(𝑀 × 𝑀)⟶𝑀)
8		isasslaw 49110	. . . . . . . 8 ⊢ (( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ 𝑀 ∈ 𝑉) → ( ⚬ assLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
9	8	biimpd 232	. . . . . . 7 ⊢ (( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ 𝑀 ∈ 𝑉) → ( ⚬ assLaw 𝑀 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
10	9	impancom 457	. . . . . 6 ⊢ (( ⚬ ∈ (𝑀 ↑m (𝑀 × 𝑀)) ∧ ⚬ assLaw 𝑀) → (𝑀 ∈ 𝑉 → ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
11	10	impcom 413	. . . . 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 521	. . . 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 418	. . 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 243	. 2 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) → ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))
15		isclintop 49125	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( clIntOp ‘𝑀) ↔ ⚬ :(𝑀 × 𝑀)⟶𝑀))
16	15	biimprcd 253	. . . . . 6 ⊢ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 → (𝑀 ∈ 𝑉 → ⚬ ∈ ( clIntOp ‘𝑀)))
17	16	adantr 486	. . . . 5 ⊢ (( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))) → (𝑀 ∈ 𝑉 → ⚬ ∈ ( clIntOp ‘𝑀)))
18	17	impcom 413	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) → ⚬ ∈ ( clIntOp ‘𝑀))
19		sqxpexg 7755	. . . . . . . . 9 ⊢ (𝑀 ∈ 𝑉 → (𝑀 × 𝑀) ∈ V)
20		fex 7226	. . . . . . . . 9 ⊢ (( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ (𝑀 × 𝑀) ∈ V) → ⚬ ∈ V)
21	19, 20	sylan2 605	. . . . . . . 8 ⊢ (( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ 𝑀 ∈ 𝑉) → ⚬ ∈ V)
22	21	ancoms 464	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ ⚬ :(𝑀 × 𝑀)⟶𝑀) → ⚬ ∈ V)
23		simpl 488	. . . . . . 7 ⊢ ((𝑀 ∈ 𝑉 ∧ ⚬ :(𝑀 × 𝑀)⟶𝑀) → 𝑀 ∈ 𝑉)
24		isasslaw 49110	. . . . . . . 8 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ 𝑉) → ( ⚬ assLaw 𝑀 ↔ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))))
25	24	bicomd 226	. . . . . . 7 ⊢ (( ⚬ ∈ V ∧ 𝑀 ∈ 𝑉) → (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ⚬ assLaw 𝑀))
26	22, 23, 25	syl2anc 596	. . . . . 6 ⊢ ((𝑀 ∈ 𝑉 ∧ ⚬ :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) ↔ ⚬ assLaw 𝑀))
27	26	biimpd 232	. . . . 5 ⊢ ((𝑀 ∈ 𝑉 ∧ ⚬ :(𝑀 × 𝑀)⟶𝑀) → (∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)) → ⚬ assLaw 𝑀))
28	27	impr 460	. . . 4 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) → ⚬ assLaw 𝑀)
29		assintopval 49123	. . . . . . 7 ⊢ (𝑀 ∈ 𝑉 → ( assIntOp ‘𝑀) = {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀})
30	29	adantr 486	. . . . . 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 2846	. . . . 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 3645	. . . . 5 ⊢ ( ⚬ ∈ {𝑜 ∈ ( clIntOp ‘𝑀) ∣ 𝑜 assLaw 𝑀} ↔ ( ⚬ ∈ ( clIntOp ‘𝑀) ∧ ⚬ assLaw 𝑀))
33	31, 32	bitrdi 290	. . . 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 726	. . 3 ⊢ ((𝑀 ∈ 𝑉 ∧ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))) → ⚬ ∈ ( assIntOp ‘𝑀))
35	34	ex 418	. 2 ⊢ (𝑀 ∈ 𝑉 → (( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧))) → ⚬ ∈ ( assIntOp ‘𝑀)))
36	14, 35	impbid 215	1 ⊢ (𝑀 ∈ 𝑉 → ( ⚬ ∈ ( assIntOp ‘𝑀) ↔ ( ⚬ :(𝑀 × 𝑀)⟶𝑀 ∧ ∀𝑥 ∈ 𝑀 ∀𝑦 ∈ 𝑀 ∀𝑧 ∈ 𝑀 ((𝑥 ⚬ 𝑦) ⚬ 𝑧) = (𝑥 ⚬ (𝑦 ⚬ 𝑧)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  {crab 3412  Vcvv 3450   class class class wbr 5103   × cxp 5653  ⟶wf 6529  ‘cfv 6533  (class class class)co 7414   ↑_m cmap 8829   assLaw casslaw 49102   clIntOp cclintop 49115   assIntOp cassintop 49116

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-map 8831  df-asslaw 49106  df-intop 49117  df-clintop 49118  df-assintop 49119

This theorem is used by: (None)