Theorem ablprop 13842

Description: If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)

Hypotheses

Ref	Expression
ablprop.b	⊢ (Base‘𝐾) = (Base‘𝐿)
ablprop.p	⊢ (+g‘𝐾) = (+g‘𝐿)

Assertion

Ref	Expression
ablprop	⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Proof of Theorem ablprop

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2230	. . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐾))
2		ablprop.b	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐿)
3	2	a1i 9	. . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐿))
4		ablprop.p	. . . . 5 ⊢ (+g‘𝐾) = (+g‘𝐿)
5	4	oveqi 6020	. . . 4 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)
6	5	a1i 9	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7	1, 3, 6	ablpropd 13841	. 2 ⊢ (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
8	7	mptru 1404	1 ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1395  ⊤wtru 1396   ∈ wcel 2200  ‘cfv 5318  (class class class)co 6007  Basecbs 13040  +_gcplusg 13118  Abelcabl 13830

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-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8098  ax-resscn 8099  ax-1re 8101  ax-addrcl 8104

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5960  df-ov 6010  df-inn 9119  df-2 9177  df-ndx 13043  df-slot 13044  df-base 13046  df-plusg 13131  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544  df-cmn 13831  df-abl 13832

This theorem is referenced by: (None)