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Theorem issubm 18763
Description: Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
Hypotheses
Ref Expression
issubm.b 𝐵 = (Base‘𝑀)
issubm.z 0 = (0g𝑀)
issubm.p + = (+g𝑀)
Assertion
Ref Expression
issubm (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   + (𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem issubm
Dummy variables 𝑚 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6896 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
21pweqd 4621 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3 fveq2 6896 . . . . . . 7 (𝑚 = 𝑀 → (0g𝑚) = (0g𝑀))
43eleq1d 2810 . . . . . 6 (𝑚 = 𝑀 → ((0g𝑚) ∈ 𝑡 ↔ (0g𝑀) ∈ 𝑡))
5 fveq2 6896 . . . . . . . . 9 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
65oveqd 7436 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥(+g𝑀)𝑦))
76eleq1d 2810 . . . . . . 7 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑡))
872ralbidv 3208 . . . . . 6 (𝑚 = 𝑀 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡))
94, 8anbi12d 630 . . . . 5 (𝑚 = 𝑀 → (((0g𝑚) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡) ↔ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)))
102, 9rabeqbidv 3436 . . . 4 (𝑚 = 𝑀 → {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ((0g𝑚) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡)} = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)})
11 df-submnd 18744 . . . 4 SubMnd = (𝑚 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ((0g𝑚) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡)})
12 fvex 6909 . . . . . 6 (Base‘𝑀) ∈ V
1312pwex 5380 . . . . 5 𝒫 (Base‘𝑀) ∈ V
1413rabex 5335 . . . 4 {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)} ∈ V
1510, 11, 14fvmpt 7004 . . 3 (𝑀 ∈ Mnd → (SubMnd‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)})
1615eleq2d 2811 . 2 (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)}))
17 eleq2 2814 . . . . 5 (𝑡 = 𝑆 → ((0g𝑀) ∈ 𝑡 ↔ (0g𝑀) ∈ 𝑆))
18 eleq2 2814 . . . . . . 7 (𝑡 = 𝑆 → ((𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1918raleqbi1dv 3322 . . . . . 6 (𝑡 = 𝑆 → (∀𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2019raleqbi1dv 3322 . . . . 5 (𝑡 = 𝑆 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2117, 20anbi12d 630 . . . 4 (𝑡 = 𝑆 → (((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡) ↔ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
2221elrab 3679 . . 3 (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)} ↔ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
23 issubm.b . . . . . 6 𝐵 = (Base‘𝑀)
2423sseq2i 4006 . . . . 5 (𝑆𝐵𝑆 ⊆ (Base‘𝑀))
25 issubm.z . . . . . . 7 0 = (0g𝑀)
2625eleq1i 2816 . . . . . 6 ( 0𝑆 ↔ (0g𝑀) ∈ 𝑆)
27 issubm.p . . . . . . . . 9 + = (+g𝑀)
2827oveqi 7432 . . . . . . . 8 (𝑥 + 𝑦) = (𝑥(+g𝑀)𝑦)
2928eleq1i 2816 . . . . . . 7 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆)
30292ralbii 3117 . . . . . 6 (∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)
3126, 30anbi12i 626 . . . . 5 (( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
3224, 31anbi12i 626 . . . 4 ((𝑆𝐵 ∧ ( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
33 3anass 1092 . . . 4 ((𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆𝐵 ∧ ( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
3412elpw2 5348 . . . . 5 (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
3534anbi1i 622 . . . 4 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
3632, 33, 353bitr4ri 303 . . 3 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
3722, 36bitri 274 . 2 (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)} ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
3816, 37bitrdi 286 1 (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  wral 3050  {crab 3418  wss 3944  𝒫 cpw 4604  cfv 6549  (class class class)co 7419  Basecbs 17183  +gcplusg 17236  0gc0g 17424  Mndcmnd 18697  SubMndcsubmnd 18742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-ov 7422  df-submnd 18744
This theorem is referenced by:  issubm2  18764  issubmd  18766  mndissubm  18767  submcl  18772  0subm  18777  insubm  18778  mhmima  18785  mhmeql  18786  submacs  18787  gsumwspan  18806  frmdsssubm  18821  sursubmefmnd  18856  injsubmefmnd  18857  issubg3  19107  cycsubm  19165  cntzsubm  19301  oppgsubm  19328  lsmsubm  19620  issubrg3  20551  isdomn3  21265  xrge0subm  21357  cnsubmlem  21364  nn0srg  21387  rge0srg  21388  efsubm  26530  rrgsubm  33072  1arithufdlem4  33362  iistmd  33634  mon1psubm  42769
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