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Theorem issubm 17540
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 6344 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
21pweqd 4299 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3 fveq2 6344 . . . . . . 7 (𝑚 = 𝑀 → (0g𝑚) = (0g𝑀))
43eleq1d 2816 . . . . . 6 (𝑚 = 𝑀 → ((0g𝑚) ∈ 𝑡 ↔ (0g𝑀) ∈ 𝑡))
5 fveq2 6344 . . . . . . . . 9 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
65oveqd 6822 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥(+g𝑀)𝑦))
76eleq1d 2816 . . . . . . 7 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑡))
872ralbidv 3119 . . . . . 6 (𝑚 = 𝑀 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡))
94, 8anbi12d 749 . . . . 5 (𝑚 = 𝑀 → (((0g𝑚) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡) ↔ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)))
102, 9rabeqbidv 3327 . . . 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 17529 . . . 4 SubMnd = (𝑚 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ((0g𝑚) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡)})
12 fvex 6354 . . . . . 6 (Base‘𝑀) ∈ V
1312pwex 4989 . . . . 5 𝒫 (Base‘𝑀) ∈ V
1413rabex 4956 . . . 4 {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)} ∈ V
1510, 11, 14fvmpt 6436 . . 3 (𝑀 ∈ Mnd → (SubMnd‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)})
1615eleq2d 2817 . 2 (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)}))
17 eleq2 2820 . . . . 5 (𝑡 = 𝑆 → ((0g𝑀) ∈ 𝑡 ↔ (0g𝑀) ∈ 𝑆))
18 eleq2 2820 . . . . . . 7 (𝑡 = 𝑆 → ((𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1918raleqbi1dv 3277 . . . . . 6 (𝑡 = 𝑆 → (∀𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2019raleqbi1dv 3277 . . . . 5 (𝑡 = 𝑆 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2117, 20anbi12d 749 . . . 4 (𝑡 = 𝑆 → (((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡) ↔ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
2221elrab 3496 . . 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 3763 . . . . 5 (𝑆𝐵𝑆 ⊆ (Base‘𝑀))
25 issubm.z . . . . . . 7 0 = (0g𝑀)
2625eleq1i 2822 . . . . . 6 ( 0𝑆 ↔ (0g𝑀) ∈ 𝑆)
27 issubm.p . . . . . . . . 9 + = (+g𝑀)
2827oveqi 6818 . . . . . . . 8 (𝑥 + 𝑦) = (𝑥(+g𝑀)𝑦)
2928eleq1i 2822 . . . . . . 7 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆)
30292ralbii 3111 . . . . . 6 (∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)
3126, 30anbi12i 735 . . . . 5 (( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
3224, 31anbi12i 735 . . . 4 ((𝑆𝐵 ∧ ( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
33 3anass 1081 . . . 4 ((𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆𝐵 ∧ ( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
3412elpw2 4969 . . . . 5 (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
3534anbi1i 733 . . . 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 293 . . 3 ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
3722, 36bitri 264 . 2 (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)} ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆))
3816, 37syl6bb 276 1 (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1624  wcel 2131  wral 3042  {crab 3046  wss 3707  𝒫 cpw 4294  cfv 6041  (class class class)co 6805  Basecbs 16051  +gcplusg 16135  0gc0g 16294  Mndcmnd 17487  SubMndcsubmnd 17527
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-submnd 17529
This theorem is referenced by:  issubm2  17541  issubmd  17542  submcl  17546  mhmima  17556  mhmeql  17557  submacs  17558  gsumwspan  17576  frmdsssubm  17591  issubg3  17805  cntzsubm  17960  oppgsubm  17984  lsmsubm  18260  issubrg3  19002  xrge0subm  19981  cnsubmlem  19988  nn0srg  20010  rge0srg  20011  efsubm  24488  iistmd  30249  isdomn3  38276  mon1psubm  38278
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