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Theorem issubmgm 44396
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 6649 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
21pweqd 4519 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3 fveq2 6649 . . . . . . . 8 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
43oveqd 7156 . . . . . . 7 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥(+g𝑀)𝑦))
54eleq1d 2877 . . . . . 6 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑡))
652ralbidv 3167 . . . . 5 (𝑚 = 𝑀 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡))
72, 6rabeqbidv 3436 . . . 4 (𝑚 = 𝑀 → {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡} = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡})
8 df-submgm 44387 . . . 4 SubMgm = (𝑚 ∈ Mgm ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡})
9 fvex 6662 . . . . . 6 (Base‘𝑀) ∈ V
109pwex 5249 . . . . 5 𝒫 (Base‘𝑀) ∈ V
1110rabex 5202 . . . 4 {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡} ∈ V
127, 8, 11fvmpt 6749 . . 3 (𝑀 ∈ Mgm → (SubMgm‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡})
1312eleq2d 2878 . 2 (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡}))
149elpw2 5215 . . . 4 (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
1514anbi1i 626 . . 3 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
16 eleq2 2881 . . . . . 6 (𝑡 = 𝑆 → ((𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1716raleqbi1dv 3359 . . . . 5 (𝑡 = 𝑆 → (∀𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1817raleqbi1dv 3359 . . . 4 (𝑡 = 𝑆 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1918elrab 3631 . . 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 3947 . . . 4 (𝑆𝐵𝑆 ⊆ (Base‘𝑀))
22 issubmgm.p . . . . . . 7 + = (+g𝑀)
2322oveqi 7152 . . . . . 6 (𝑥 + 𝑦) = (𝑥(+g𝑀)𝑦)
2423eleq1i 2883 . . . . 5 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆)
25242ralbii 3137 . . . 4 (∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)
2621, 25anbi12i 629 . . 3 ((𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2715, 19, 263bitr4i 306 . 2 (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡} ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
2813, 27syl6bb 290 1 (𝑀 ∈ Mgm → (𝑆 ∈ (SubMgm‘𝑀) ↔ (𝑆𝐵 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wral 3109  {crab 3113  wss 3884  𝒫 cpw 4500  cfv 6328  (class class class)co 7139  Basecbs 16478  +gcplusg 16560  Mgmcmgm 17845  SubMgmcsubmgm 44385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-submgm 44387
This theorem is referenced by:  issubmgm2  44397  rabsubmgmd  44398  submgmcl  44401  mgmhmima  44409  mgmhmeql  44410  submgmacs  44411
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