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Theorem issubm 17614
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 6374 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
21pweqd 4319 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3 fveq2 6374 . . . . . . 7 (𝑚 = 𝑀 → (0g𝑚) = (0g𝑀))
43eleq1d 2828 . . . . . 6 (𝑚 = 𝑀 → ((0g𝑚) ∈ 𝑡 ↔ (0g𝑀) ∈ 𝑡))
5 fveq2 6374 . . . . . . . . 9 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
65oveqd 6858 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥(+g𝑀)𝑦))
76eleq1d 2828 . . . . . . 7 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑡))
872ralbidv 3135 . . . . . 6 (𝑚 = 𝑀 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡))
94, 8anbi12d 624 . . . . 5 (𝑚 = 𝑀 → (((0g𝑚) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡) ↔ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)))
102, 9rabeqbidv 3343 . . . 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 17603 . . . 4 SubMnd = (𝑚 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ((0g𝑚) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡)})
12 fvex 6387 . . . . . 6 (Base‘𝑀) ∈ V
1312pwex 5015 . . . . 5 𝒫 (Base‘𝑀) ∈ V
1413rabex 4972 . . . 4 {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)} ∈ V
1510, 11, 14fvmpt 6470 . . 3 (𝑀 ∈ Mnd → (SubMnd‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)})
1615eleq2d 2829 . 2 (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)}))
17 eleq2 2832 . . . . 5 (𝑡 = 𝑆 → ((0g𝑀) ∈ 𝑡 ↔ (0g𝑀) ∈ 𝑆))
18 eleq2 2832 . . . . . . 7 (𝑡 = 𝑆 → ((𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1918raleqbi1dv 3293 . . . . . 6 (𝑡 = 𝑆 → (∀𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2019raleqbi1dv 3293 . . . . 5 (𝑡 = 𝑆 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2117, 20anbi12d 624 . . . 4 (𝑡 = 𝑆 → (((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡) ↔ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
2221elrab 3518 . . 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 3789 . . . . 5 (𝑆𝐵𝑆 ⊆ (Base‘𝑀))
25 issubm.z . . . . . . 7 0 = (0g𝑀)
2625eleq1i 2834 . . . . . 6 ( 0𝑆 ↔ (0g𝑀) ∈ 𝑆)
27 issubm.p . . . . . . . . 9 + = (+g𝑀)
2827oveqi 6854 . . . . . . . 8 (𝑥 + 𝑦) = (𝑥(+g𝑀)𝑦)
2928eleq1i 2834 . . . . . . 7 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆)
30292ralbii 3127 . . . . . 6 (∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)
3126, 30anbi12i 620 . . . . 5 (( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
3224, 31anbi12i 620 . . . 4 ((𝑆𝐵 ∧ ( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
33 3anass 1116 . . . 4 ((𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆𝐵 ∧ ( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
3412elpw2 4985 . . . . 5 (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
3534anbi1i 617 . . . 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 295 . . 3 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
3722, 36bitri 266 . 2 (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)} ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
3816, 37syl6bb 278 1 (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wral 3054  {crab 3058  wss 3731  𝒫 cpw 4314  cfv 6067  (class class class)co 6841  Basecbs 16131  +gcplusg 16215  0gc0g 16367  Mndcmnd 17561  SubMndcsubmnd 17601
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 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061
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 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3351  df-sbc 3596  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-op 4340  df-uni 4594  df-br 4809  df-opab 4871  df-mpt 4888  df-id 5184  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-iota 6030  df-fun 6069  df-fv 6075  df-ov 6844  df-submnd 17603
This theorem is referenced by:  issubm2  17615  issubmd  17616  submcl  17620  mhmima  17630  mhmeql  17631  submacs  17632  gsumwspan  17651  frmdsssubm  17666  issubg3  17877  cntzsubm  18032  oppgsubm  18056  lsmsubm  18333  issubrg3  19076  xrge0subm  20059  cnsubmlem  20066  nn0srg  20088  rge0srg  20089  efsubm  24588  iistmd  30329  isdomn3  38391  mon1psubm  38393
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