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Theorem issubmgm 18715
Description: Expand definition of a submagma. (Contributed by AV, 25-Feb-2020.)
Hypotheses
Ref Expression
issubmgm.b 𝐵 = (Base‘𝑀)
issubmgm.p + = (+g𝑀)
Assertion
Ref Expression
issubmgm (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   + (𝑥,𝑦)

Proof of Theorem issubmgm
Dummy variables 𝑚 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6906 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
21pweqd 4617 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3 fveq2 6906 . . . . . . . 8 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
43oveqd 7448 . . . . . . 7 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥(+g𝑀)𝑦))
54eleq1d 2826 . . . . . 6 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑡))
652ralbidv 3221 . . . . 5 (𝑚 = 𝑀 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡))
72, 6rabeqbidv 3455 . . . 4 (𝑚 = 𝑀 → {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡} = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡})
8 df-submgm 18706 . . . 4 SubMgm = (𝑚 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡})
9 fvex 6919 . . . . . 6 (Base‘𝑀) ∈ V
109pwex 5380 . . . . 5 𝒫 (Base‘𝑀) ∈ V
1110rabex 5339 . . . 4 {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡} ∈ V
127, 8, 11fvmpt 7016 . . 3 (𝑀 ∈ Mgm → (SubMgm‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡})
1312eleq2d 2827 . 2 (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡}))
149elpw2 5334 . . . 4 (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
1514anbi1i 624 . . 3 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
16 eleq2 2830 . . . . . 6 (𝑡 = 𝑆 → ((𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1716raleqbi1dv 3338 . . . . 5 (𝑡 = 𝑆 → (∀𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1817raleqbi1dv 3338 . . . 4 (𝑡 = 𝑆 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1918elrab 3692 . . 3 (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡} ↔ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
20 issubmgm.b . . . . 5 𝐵 = (Base‘𝑀)
2120sseq2i 4013 . . . 4 (𝑆𝐵𝑆 ⊆ (Base‘𝑀))
22 issubmgm.p . . . . . . 7 + = (+g𝑀)
2322oveqi 7444 . . . . . 6 (𝑥 + 𝑦) = (𝑥(+g𝑀)𝑦)
2423eleq1i 2832 . . . . 5 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆)
25242ralbii 3128 . . . 4 (∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)
2621, 25anbi12i 628 . . 3 ((𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2715, 19, 263bitr4i 303 . 2 (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡} ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
2813, 27bitrdi 287 1 (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  {crab 3436  wss 3951  𝒫 cpw 4600  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Mgmcmgm 18651  SubMgmcsubmgm 18704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-submgm 18706
This theorem is referenced by:  issubmgm2  18716  rabsubmgmd  18717  submgmcl  18720  mgmhmima  18728  mgmhmeql  18729  submgmacs  18730
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