grpprop - Metamath Proof Explorer

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Theorem grpprop 18576

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

**Hypotheses**
Ref	Expression
grpprop.b	⊢ (Base‘𝐾) = (Base‘𝐿)
grpprop.p	⊢ (+_g‘𝐾) = (+_g‘𝐿)

**Assertion**
Ref	Expression
grpprop	⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Proof of Theorem grpprop Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2740	. . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐾))
2		grpprop.b	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐿)
3	2	a1i 11	. . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐿))
4		grpprop.p	. . . . 5 ⊢ (+_g‘𝐾) = (+_g‘𝐿)
5	4	oveqi 7281	. . . 4 ⊢ (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦)
6	5	a1i 11	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦))
7	1, 3, 6	grppropd 18575	. 2 ⊢ (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
8	7	mptru 1548	1 ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1541 ⊤wtru 1542 ∈ wcel 2109 ‘cfv 6430 (class class class)co 7268 Basecbs 16893 +_gcplusg 16943 Grpcgrp 18558

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1801 ax-4 1815 ax-5 1916 ax-6 1974 ax-7 2014 ax-8 2111 ax-9 2119 ax-10 2140 ax-11 2157 ax-12 2174 ax-ext 2710 ax-sep 5226 ax-nul 5233 ax-pr 5355

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1544 df-fal 1554 df-ex 1786 df-nf 1790 df-sb 2071 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-ral 3070 df-rex 3071 df-rab 3074 df-v 3432 df-sbc 3720 df-dif 3894 df-un 3896 df-in 3898 df-ss 3908 df-nul 4262 df-if 4465 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4845 df-br 5079 df-opab 5141 df-mpt 5162 df-id 5488 df-xp 5594 df-rel 5595 df-cnv 5596 df-co 5597 df-dm 5598 df-iota 6388 df-fun 6432 df-fv 6438 df-ov 7271 df-0g 17133 df-mgm 18307 df-sgrp 18356 df-mnd 18367 df-grp 18561

This theorem is referenced by: grppropstr 18577 grpss 18578 opprring 19854 opprsubg 19859 rmodislmod 20172 rmodislmodOLD 20173 lmod1 45785

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