Theorem ismgmOLD 35143

Description: Obsolete version of ismgm 17853 as of 3-Feb-2020. The predicate "is a magma". (Contributed by FL, 2-Nov-2009.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
ismgmOLD.1	⊢ 𝑋 = dom dom 𝐺

Assertion

Ref	Expression
ismgmOLD	⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))

Proof of Theorem ismgmOLD

Dummy variables 𝑔 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		feq1 6495	. . . . 5 ⊢ (𝑔 = 𝐺 → (𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑡 × 𝑡)⟶𝑡))
2	1	exbidv 1922	. . . 4 ⊢ (𝑔 = 𝐺 → (∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
3		df-mgmOLD 35142	. . . 4 ⊢ Magma = {𝑔 ∣ ∃𝑡 𝑔:(𝑡 × 𝑡)⟶𝑡}
4	2, 3	elab2g 3668	. . 3 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡))
5		f00 6561	. . . . . . . 8 ⊢ (𝐺:(∅ × ∅)⟶∅ ↔ (𝐺 = ∅ ∧ (∅ × ∅) = ∅))
6		dmeq 5772	. . . . . . . . . 10 ⊢ (𝐺 = ∅ → dom 𝐺 = dom ∅)
7		dmeq 5772	. . . . . . . . . . 11 ⊢ (dom 𝐺 = dom ∅ → dom dom 𝐺 = dom dom ∅)
8		dm0 5790	. . . . . . . . . . . . 13 ⊢ dom ∅ = ∅
9	8	dmeqi 5773	. . . . . . . . . . . 12 ⊢ dom dom ∅ = dom ∅
10	9, 8	eqtri 2844	. . . . . . . . . . 11 ⊢ dom dom ∅ = ∅
11	7, 10	syl6req 2873	. . . . . . . . . 10 ⊢ (dom 𝐺 = dom ∅ → ∅ = dom dom 𝐺)
12	6, 11	syl 17	. . . . . . . . 9 ⊢ (𝐺 = ∅ → ∅ = dom dom 𝐺)
13	12	adantr 483	. . . . . . . 8 ⊢ ((𝐺 = ∅ ∧ (∅ × ∅) = ∅) → ∅ = dom dom 𝐺)
14	5, 13	sylbi 219	. . . . . . 7 ⊢ (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)
15		xpeq12 5580	. . . . . . . . . 10 ⊢ ((𝑡 = ∅ ∧ 𝑡 = ∅) → (𝑡 × 𝑡) = (∅ × ∅))
16	15	anidms 569	. . . . . . . . 9 ⊢ (𝑡 = ∅ → (𝑡 × 𝑡) = (∅ × ∅))
17		feq23 6498	. . . . . . . . 9 ⊢ (((𝑡 × 𝑡) = (∅ × ∅) ∧ 𝑡 = ∅) → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(∅ × ∅)⟶∅))
18	16, 17	mpancom 686	. . . . . . . 8 ⊢ (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(∅ × ∅)⟶∅))
19		eqeq1 2825	. . . . . . . 8 ⊢ (𝑡 = ∅ → (𝑡 = dom dom 𝐺 ↔ ∅ = dom dom 𝐺))
20	18, 19	imbi12d 347	. . . . . . 7 ⊢ (𝑡 = ∅ → ((𝐺:(𝑡 × 𝑡)⟶𝑡 → 𝑡 = dom dom 𝐺) ↔ (𝐺:(∅ × ∅)⟶∅ → ∅ = dom dom 𝐺)))
21	14, 20	mpbiri 260	. . . . . 6 ⊢ (𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡 → 𝑡 = dom dom 𝐺))
22		fdm 6522	. . . . . . . 8 ⊢ (𝐺:(𝑡 × 𝑡)⟶𝑡 → dom 𝐺 = (𝑡 × 𝑡))
23		dmeq 5772	. . . . . . . 8 ⊢ (dom 𝐺 = (𝑡 × 𝑡) → dom dom 𝐺 = dom (𝑡 × 𝑡))
24		df-ne 3017	. . . . . . . . . . . 12 ⊢ (𝑡 ≠ ∅ ↔ ¬ 𝑡 = ∅)
25		dmxp 5799	. . . . . . . . . . . 12 ⊢ (𝑡 ≠ ∅ → dom (𝑡 × 𝑡) = 𝑡)
26	24, 25	sylbir 237	. . . . . . . . . . 11 ⊢ (¬ 𝑡 = ∅ → dom (𝑡 × 𝑡) = 𝑡)
27	26	eqeq1d 2823	. . . . . . . . . 10 ⊢ (¬ 𝑡 = ∅ → (dom (𝑡 × 𝑡) = dom dom 𝐺 ↔ 𝑡 = dom dom 𝐺))
28	27	biimpcd 251	. . . . . . . . 9 ⊢ (dom (𝑡 × 𝑡) = dom dom 𝐺 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
29	28	eqcoms 2829	. . . . . . . 8 ⊢ (dom dom 𝐺 = dom (𝑡 × 𝑡) → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
30	22, 23, 29	3syl 18	. . . . . . 7 ⊢ (𝐺:(𝑡 × 𝑡)⟶𝑡 → (¬ 𝑡 = ∅ → 𝑡 = dom dom 𝐺))
31	30	com12 32	. . . . . 6 ⊢ (¬ 𝑡 = ∅ → (𝐺:(𝑡 × 𝑡)⟶𝑡 → 𝑡 = dom dom 𝐺))
32	21, 31	pm2.61i 184	. . . . 5 ⊢ (𝐺:(𝑡 × 𝑡)⟶𝑡 → 𝑡 = dom dom 𝐺)
33	32	pm4.71ri 563	. . . 4 ⊢ (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ (𝑡 = dom dom 𝐺 ∧ 𝐺:(𝑡 × 𝑡)⟶𝑡))
34	33	exbii 1848	. . 3 ⊢ (∃𝑡 𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ ∃𝑡(𝑡 = dom dom 𝐺 ∧ 𝐺:(𝑡 × 𝑡)⟶𝑡))
35	4, 34	syl6bb 289	. 2 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ ∃𝑡(𝑡 = dom dom 𝐺 ∧ 𝐺:(𝑡 × 𝑡)⟶𝑡)))
36		dmexg 7613	. . 3 ⊢ (𝐺 ∈ 𝐴 → dom 𝐺 ∈ V)
37		dmexg 7613	. . 3 ⊢ (dom 𝐺 ∈ V → dom dom 𝐺 ∈ V)
38		xpeq12 5580	. . . . . . 7 ⊢ ((𝑡 = dom dom 𝐺 ∧ 𝑡 = dom dom 𝐺) → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
39	38	anidms 569	. . . . . 6 ⊢ (𝑡 = dom dom 𝐺 → (𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺))
40		feq23 6498	. . . . . 6 ⊢ (((𝑡 × 𝑡) = (dom dom 𝐺 × dom dom 𝐺) ∧ 𝑡 = dom dom 𝐺) → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
41	39, 40	mpancom 686	. . . . 5 ⊢ (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺))
42		ismgmOLD.1	. . . . . . . 8 ⊢ 𝑋 = dom dom 𝐺
43	42	eqcomi 2830	. . . . . . 7 ⊢ dom dom 𝐺 = 𝑋
44	43, 43	xpeq12i 5583	. . . . . 6 ⊢ (dom dom 𝐺 × dom dom 𝐺) = (𝑋 × 𝑋)
45	44, 43	feq23i 6508	. . . . 5 ⊢ (𝐺:(dom dom 𝐺 × dom dom 𝐺)⟶dom dom 𝐺 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋)
46	41, 45	syl6bb 289	. . . 4 ⊢ (𝑡 = dom dom 𝐺 → (𝐺:(𝑡 × 𝑡)⟶𝑡 ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
47	46	ceqsexgv 3647	. . 3 ⊢ (dom dom 𝐺 ∈ V → (∃𝑡(𝑡 = dom dom 𝐺 ∧ 𝐺:(𝑡 × 𝑡)⟶𝑡) ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
48	36, 37, 47	3syl 18	. 2 ⊢ (𝐺 ∈ 𝐴 → (∃𝑡(𝑡 = dom dom 𝐺 ∧ 𝐺:(𝑡 × 𝑡)⟶𝑡) ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))
49	35, 48	bitrd 281	1 ⊢ (𝐺 ∈ 𝐴 → (𝐺 ∈ Magma ↔ 𝐺:(𝑋 × 𝑋)⟶𝑋))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114   ≠ wne 3016  Vcvv 3494  ∅c0 4291   × cxp 5553  dom cdm 5555  ⟶wf 6351  Magmacmagm 35141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-fun 6357  df-fn 6358  df-f 6359  df-mgmOLD 35142

This theorem is referenced by:  clmgmOLD  35144  opidonOLD  35145  issmgrpOLD  35156