Theorem isdrngtap 14466

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 5672	. . . . 5 ⊢ (𝑟 = 𝑅 → (#r‘𝑟) = (#r‘𝑅))
2		isdrngap.ap	. . . . 5 ⊢ # = (#r‘𝑅)
3	1, 2	eqtr4di 2285	. . . 4 ⊢ (𝑟 = 𝑅 → (#r‘𝑟) = # )
4		tapeq1 7571	. . . 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 5672	. . . . 5 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
7		isdrng.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
8	6, 7	eqtr4di 2285	. . . 4 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
9		tapeq2 7572	. . . 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 14464	. 2 ⊢ DivRing = {𝑟 ∈ Ring ∣ (#r‘𝑟) TAp (Base‘𝑟)}
13	11, 12	elrab2 2978	1 ⊢ (𝑅 ∈ DivRing ↔ (𝑅 ∈ Ring ∧ # TAp 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205  ‘cfv 5354   TAp wtap 7567  Basecbs 13233  Ringcrg 14161  #_rcapr 14449  DivRingcdr 14462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-xp 4757  df-iota 5314  df-fv 5362  df-pap 7561  df-tap 7568  df-drngap 14464

This theorem is referenced by:  drnglring  14467  drngprop  14477  opprdrng  14480