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Theorem issubm 17279
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 6153 . . . . . 6 (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
21pweqd 4140 . . . . 5 (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3 fveq2 6153 . . . . . . 7 (𝑚 = 𝑀 → (0g𝑚) = (0g𝑀))
43eleq1d 2683 . . . . . 6 (𝑚 = 𝑀 → ((0g𝑚) ∈ 𝑡 ↔ (0g𝑀) ∈ 𝑡))
5 fveq2 6153 . . . . . . . . 9 (𝑚 = 𝑀 → (+g𝑚) = (+g𝑀))
65oveqd 6627 . . . . . . . 8 (𝑚 = 𝑀 → (𝑥(+g𝑚)𝑦) = (𝑥(+g𝑀)𝑦))
76eleq1d 2683 . . . . . . 7 (𝑚 = 𝑀 → ((𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑡))
872ralbidv 2984 . . . . . 6 (𝑚 = 𝑀 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡))
94, 8anbi12d 746 . . . . 5 (𝑚 = 𝑀 → (((0g𝑚) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡) ↔ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)))
102, 9rabeqbidv 3184 . . . 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 17268 . . . 4 SubMnd = (𝑚 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ((0g𝑚) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑚)𝑦) ∈ 𝑡)})
12 fvex 6163 . . . . . 6 (Base‘𝑀) ∈ V
1312pwex 4813 . . . . 5 𝒫 (Base‘𝑀) ∈ V
1413rabex 4778 . . . 4 {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)} ∈ V
1510, 11, 14fvmpt 6244 . . 3 (𝑀 ∈ Mnd → (SubMnd‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)})
1615eleq2d 2684 . 2 (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡)}))
17 eleq2 2687 . . . . 5 (𝑡 = 𝑆 → ((0g𝑀) ∈ 𝑡 ↔ (0g𝑀) ∈ 𝑆))
18 eleq2 2687 . . . . . . 7 (𝑡 = 𝑆 → ((𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆))
1918raleqbi1dv 3138 . . . . . 6 (𝑡 = 𝑆 → (∀𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2019raleqbi1dv 3138 . . . . 5 (𝑡 = 𝑆 → (∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
2117, 20anbi12d 746 . . . 4 (𝑡 = 𝑆 → (((0g𝑀) ∈ 𝑡 ∧ ∀𝑥𝑡𝑦𝑡 (𝑥(+g𝑀)𝑦) ∈ 𝑡) ↔ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
2221elrab 3350 . . 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 3614 . . . . 5 (𝑆𝐵𝑆 ⊆ (Base‘𝑀))
25 issubm.z . . . . . . 7 0 = (0g𝑀)
2625eleq1i 2689 . . . . . 6 ( 0𝑆 ↔ (0g𝑀) ∈ 𝑆)
27 issubm.p . . . . . . . . 9 + = (+g𝑀)
2827oveqi 6623 . . . . . . . 8 (𝑥 + 𝑦) = (𝑥(+g𝑀)𝑦)
2928eleq1i 2689 . . . . . . 7 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g𝑀)𝑦) ∈ 𝑆)
30292ralbii 2976 . . . . . 6 (∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)
3126, 30anbi12i 732 . . . . 5 (( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆))
3224, 31anbi12i 732 . . . 4 ((𝑆𝐵 ∧ ( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ((0g𝑀) ∈ 𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥(+g𝑀)𝑦) ∈ 𝑆)))
33 3anass 1040 . . . 4 ((𝑆𝐵0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆𝐵 ∧ ( 0𝑆 ∧ ∀𝑥𝑆𝑦𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
3412elpw2 4793 . . . . 5 (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
3534anbi1i 730 . . . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  {crab 2911  wss 3559  𝒫 cpw 4135  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  0gc0g 16032  Mndcmnd 17226  SubMndcsubmnd 17266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-iota 5815  df-fun 5854  df-fv 5860  df-ov 6613  df-submnd 17268
This theorem is referenced by:  issubm2  17280  issubmd  17281  submcl  17285  mhmima  17295  mhmeql  17296  submacs  17297  gsumwspan  17315  frmdsssubm  17330  issubg3  17544  cntzsubm  17700  oppgsubm  17724  lsmsubm  18000  issubrg3  18740  xrge0subm  19719  cnsubmlem  19726  nn0srg  19748  rge0srg  19749  efsubm  24218  iistmd  29754  isdomn3  37298  mon1psubm  37300
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