Theorem isdrngtap 14608

Description: The predicate "is a division ring". (Contributed by Jim Kingdon, 29-May-2026.)

Hypotheses

Ref	Expression
isdrng.b	⊢ 𝐵 = (Base‘𝑅)
isdrngap.ap	⊢ # = (#r‘𝑅)

Assertion

Ref	Expression
isdrngtap	⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ # TAp 𝐵))

Proof of Theorem isdrngtap

Dummy variable 𝑟 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5695	. . . . 5 ⊢ (𝑟 = 𝑅 → (#r‘𝑟) = (#r‘𝑅))
2		isdrngap.ap	. . . . 5 ⊢ # = (#r‘𝑅)
3	1, 2	eqtr4di 2289	. . . 4 ⊢ (𝑟 = 𝑅 → (#r‘𝑟) = # )
4		tapeq1 7618	. . . 4 ⊢ ((#r‘𝑟) = # → ((#r‘𝑟) TAp (Base‘𝑟) ↔ # TAp (Base‘𝑟)))
5	3, 4	syl 14	. . 3 ⊢ (𝑟 = 𝑅 → ((#r‘𝑟) TAp (Base‘𝑟) ↔ # TAp (Base‘𝑟)))
6		fveq2 5695	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
7		isdrng.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
8	6, 7	eqtr4di 2289	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
9		tapeq2 7619	. . . 4 ⊢ ((Base‘𝑟) = 𝐵 → ( # TAp (Base‘𝑟) ↔ # TAp 𝐵))
10	8, 9	syl 14	. . 3 ⊢ (𝑟 = 𝑅 → ( # TAp (Base‘𝑟) ↔ # TAp 𝐵))
11	5, 10	bitrd 188	. 2 ⊢ (𝑟 = 𝑅 → ((#r‘𝑟) TAp (Base‘𝑟) ↔ # TAp 𝐵))
12		df-drngap 14606	. 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (#r‘𝑟) TAp (Base‘𝑟)}
13	11, 12	elrab2 2985	1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ # TAp 𝐵))

This proof depends on syntax axioms:   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209  ‘cfv 5377   TAp wtap 7614  Basecbs 13354  Ringcrg 14302  #_rcapr 14591  DivRingcdr 14604

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-opab 4193  df-xp 4780  df-iota 5337  df-fv 5385  df-pap 7608  df-tap 7615  df-drngap 14606

This theorem is used by:  drnglring  14609  drngprop  14619  opprdrng  14622