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Theorem rabsubmgmd 41581
 Description: Deduction for proving that a restricted class abstraction is a submagma. (Contributed by AV, 26-Feb-2020.)
Hypotheses
Ref Expression
rabsubmgmd.b 𝐵 = (Base‘𝑀)
rabsubmgmd.p + = (+g𝑀)
rabsubmgmd.m (𝜑𝑀 ∈ Mgm)
rabsubmgmd.cp ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏))) → 𝜂)
rabsubmgmd.th (𝑧 = 𝑥 → (𝜓𝜃))
rabsubmgmd.ta (𝑧 = 𝑦 → (𝜓𝜏))
rabsubmgmd.et (𝑧 = (𝑥 + 𝑦) → (𝜓𝜂))
Assertion
Ref Expression
rabsubmgmd (𝜑 → {𝑧𝐵𝜓} ∈ (SubMgm‘𝑀))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑀,𝑦   𝜑,𝑥,𝑦   𝜓,𝑥,𝑦   𝑧, +   𝜂,𝑧   𝜏,𝑧   𝜃,𝑧
Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑧)   𝜃(𝑥,𝑦)   𝜏(𝑥,𝑦)   𝜂(𝑥,𝑦)   + (𝑥,𝑦)   𝑀(𝑧)

Proof of Theorem rabsubmgmd
StepHypRef Expression
1 ssrab2 3650 . . 3 {𝑧𝐵𝜓} ⊆ 𝐵
21a1i 11 . 2 (𝜑 → {𝑧𝐵𝜓} ⊆ 𝐵)
3 rabsubmgmd.th . . . . . 6 (𝑧 = 𝑥 → (𝜓𝜃))
43elrab 3331 . . . . 5 (𝑥 ∈ {𝑧𝐵𝜓} ↔ (𝑥𝐵𝜃))
5 rabsubmgmd.ta . . . . . 6 (𝑧 = 𝑦 → (𝜓𝜏))
65elrab 3331 . . . . 5 (𝑦 ∈ {𝑧𝐵𝜓} ↔ (𝑦𝐵𝜏))
74, 6anbi12i 729 . . . 4 ((𝑥 ∈ {𝑧𝐵𝜓} ∧ 𝑦 ∈ {𝑧𝐵𝜓}) ↔ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏)))
8 rabsubmgmd.m . . . . . . 7 (𝜑𝑀 ∈ Mgm)
98adantr 480 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → 𝑀 ∈ Mgm)
10 simprll 798 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → 𝑥𝐵)
11 simprrl 800 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → 𝑦𝐵)
12 rabsubmgmd.b . . . . . . 7 𝐵 = (Base‘𝑀)
13 rabsubmgmd.p . . . . . . 7 + = (+g𝑀)
1412, 13mgmcl 17068 . . . . . 6 ((𝑀 ∈ Mgm ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
159, 10, 11, 14syl3anc 1318 . . . . 5 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → (𝑥 + 𝑦) ∈ 𝐵)
16 simpl 472 . . . . . . . 8 ((𝑥𝐵𝜃) → 𝑥𝐵)
17 simpl 472 . . . . . . . 8 ((𝑦𝐵𝜏) → 𝑦𝐵)
1816, 17anim12i 588 . . . . . . 7 (((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏)) → (𝑥𝐵𝑦𝐵))
19 simpr 476 . . . . . . . 8 ((𝑥𝐵𝜃) → 𝜃)
20 simpr 476 . . . . . . . 8 ((𝑦𝐵𝜏) → 𝜏)
2119, 20anim12i 588 . . . . . . 7 (((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏)) → (𝜃𝜏))
2218, 21jca 553 . . . . . 6 (((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏)) → ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏)))
23 rabsubmgmd.cp . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏))) → 𝜂)
2422, 23sylan2 490 . . . . 5 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → 𝜂)
25 rabsubmgmd.et . . . . . 6 (𝑧 = (𝑥 + 𝑦) → (𝜓𝜂))
2625elrab 3331 . . . . 5 ((𝑥 + 𝑦) ∈ {𝑧𝐵𝜓} ↔ ((𝑥 + 𝑦) ∈ 𝐵𝜂))
2715, 24, 26sylanbrc 695 . . . 4 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → (𝑥 + 𝑦) ∈ {𝑧𝐵𝜓})
287, 27sylan2b 491 . . 3 ((𝜑 ∧ (𝑥 ∈ {𝑧𝐵𝜓} ∧ 𝑦 ∈ {𝑧𝐵𝜓})) → (𝑥 + 𝑦) ∈ {𝑧𝐵𝜓})
2928ralrimivva 2954 . 2 (𝜑 → ∀𝑥 ∈ {𝑧𝐵𝜓}∀𝑦 ∈ {𝑧𝐵𝜓} (𝑥 + 𝑦) ∈ {𝑧𝐵𝜓})
3012, 13issubmgm 41579 . . 3 (𝑀 ∈ Mgm → ({𝑧𝐵𝜓} ∈ (SubMgm‘𝑀) ↔ ({𝑧𝐵𝜓} ⊆ 𝐵 ∧ ∀𝑥 ∈ {𝑧𝐵𝜓}∀𝑦 ∈ {𝑧𝐵𝜓} (𝑥 + 𝑦) ∈ {𝑧𝐵𝜓})))
318, 30syl 17 . 2 (𝜑 → ({𝑧𝐵𝜓} ∈ (SubMgm‘𝑀) ↔ ({𝑧𝐵𝜓} ⊆ 𝐵 ∧ ∀𝑥 ∈ {𝑧𝐵𝜓}∀𝑦 ∈ {𝑧𝐵𝜓} (𝑥 + 𝑦) ∈ {𝑧𝐵𝜓})))
322, 29, 31mpbir2and 959 1 (𝜑 → {𝑧𝐵𝜓} ∈ (SubMgm‘𝑀))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900   ⊆ wss 3540  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Mgmcmgm 17063  SubMgmcsubmgm 41568 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-mgm 17065  df-submgm 41570 This theorem is referenced by: (None)
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