Theorem ablprop 13800

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 2210	. . 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 5987	. . . 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 13799	. 2 ⊢ (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
8	7	mptru 1384	1 ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1375  ⊤wtru 1376   ∈ wcel 2180  ‘cfv 5294  (class class class)co 5974  Basecbs 12998  +_gcplusg 13076  Abelcabl 13788

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-cnex 8058  ax-resscn 8059  ax-1re 8061  ax-addrcl 8064

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-int 3903  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-iota 5254  df-fun 5296  df-fn 5297  df-fv 5302  df-riota 5927  df-ov 5977  df-inn 9079  df-2 9137  df-ndx 13001  df-slot 13002  df-base 13004  df-plusg 13089  df-0g 13257  df-mgm 13355  df-sgrp 13401  df-mnd 13416  df-grp 13502  df-cmn 13789  df-abl 13790

This theorem is referenced by: (None)