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Theorem issubmgm 42390
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 6375 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
21pweqd 4320 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3 fveq2 6375 . . . . . . . 8 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
43oveqd 6859 . . . . . . 7 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥(+g𝑀)𝑦))
54eleq1d 2829 . . . . . 6 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑡))
652ralbidv 3136 . . . . 5 (𝑚 = 𝑀 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡))
72, 6rabeqbidv 3344 . . . 4 (𝑚 = 𝑀 → {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡} = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡})
8 df-submgm 42381 . . . 4 SubMgm = (𝑚 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡})
9 fvex 6388 . . . . . 6 (Base‘𝑀) ∈ V
109pwex 5016 . . . . 5 𝒫 (Base‘𝑀) ∈ V
1110rabex 4973 . . . 4 {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡} ∈ V
127, 8, 11fvmpt 6471 . . 3 (𝑀 ∈ Mgm → (SubMgm‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡})
1312eleq2d 2830 . 2 (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡}))
149elpw2 4986 . . . 4 (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
1514anbi1i 617 . . 3 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
16 eleq2 2833 . . . . . 6 (𝑡 = 𝑆 → ((𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1716raleqbi1dv 3294 . . . . 5 (𝑡 = 𝑆 → (∀𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1817raleqbi1dv 3294 . . . 4 (𝑡 = 𝑆 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1918elrab 3519 . . 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 3790 . . . 4 (𝑆𝐵𝑆 ⊆ (Base‘𝑀))
22 issubmgm.p . . . . . . 7 + = (+g𝑀)
2322oveqi 6855 . . . . . 6 (𝑥 + 𝑦) = (𝑥(+g𝑀)𝑦)
2423eleq1i 2835 . . . . 5 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆)
25242ralbii 3128 . . . 4 (∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)
2621, 25anbi12i 620 . . 3 ((𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2715, 19, 263bitr4i 294 . 2 (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡} ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
2813, 27syl6bb 278 1 (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059  wss 3732  𝒫 cpw 4315  cfv 6068  (class class class)co 6842  Basecbs 16132  +gcplusg 16216  Mgmcmgm 17508  SubMgmcsubmgm 42379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-submgm 42381
This theorem is referenced by:  issubmgm2  42391  rabsubmgmd  42392  submgmcl  42395  mgmhmima  42403  mgmhmeql  42404  submgmacs  42405
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