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Theorem issubmd 17330
Description: Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
issubmd.b 𝐵 = (Base‘𝑀)
issubmd.p + = (+g𝑀)
issubmd.z 0 = (0g𝑀)
issubmd.m (𝜑𝑀 ∈ Mnd)
issubmd.cz (𝜑𝜒)
issubmd.cp ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏))) → 𝜂)
issubmd.ch (𝑧 = 0 → (𝜓𝜒))
issubmd.th (𝑧 = 𝑥 → (𝜓𝜃))
issubmd.ta (𝑧 = 𝑦 → (𝜓𝜏))
issubmd.et (𝑧 = (𝑥 + 𝑦) → (𝜓𝜂))
Assertion
Ref Expression
issubmd (𝜑 → {𝑧𝐵𝜓} ∈ (SubMnd‘𝑀))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐵   𝑥,𝑀,𝑦   𝜑,𝑥,𝑦   𝜓,𝑥,𝑦   𝑧, +   𝑧, 0   𝜒,𝑧   𝜂,𝑧   𝜏,𝑧   𝜃,𝑧
Allowed substitution hints:   𝜑(𝑧)   𝜓(𝑧)   𝜒(𝑥,𝑦)   𝜃(𝑥,𝑦)   𝜏(𝑥,𝑦)   𝜂(𝑥,𝑦)   + (𝑥,𝑦)   𝑀(𝑧)   0 (𝑥,𝑦)

Proof of Theorem issubmd
StepHypRef Expression
1 ssrab2 3679 . . 3 {𝑧𝐵𝜓} ⊆ 𝐵
21a1i 11 . 2 (𝜑 → {𝑧𝐵𝜓} ⊆ 𝐵)
3 issubmd.m . . . 4 (𝜑𝑀 ∈ Mnd)
4 issubmd.b . . . . 5 𝐵 = (Base‘𝑀)
5 issubmd.z . . . . 5 0 = (0g𝑀)
64, 5mndidcl 17289 . . . 4 (𝑀 ∈ Mnd → 0𝐵)
73, 6syl 17 . . 3 (𝜑0𝐵)
8 issubmd.cz . . 3 (𝜑𝜒)
9 issubmd.ch . . . 4 (𝑧 = 0 → (𝜓𝜒))
109elrab 3357 . . 3 ( 0 ∈ {𝑧𝐵𝜓} ↔ ( 0𝐵𝜒))
117, 8, 10sylanbrc 697 . 2 (𝜑0 ∈ {𝑧𝐵𝜓})
12 issubmd.th . . . . . 6 (𝑧 = 𝑥 → (𝜓𝜃))
1312elrab 3357 . . . . 5 (𝑥 ∈ {𝑧𝐵𝜓} ↔ (𝑥𝐵𝜃))
14 issubmd.ta . . . . . 6 (𝑧 = 𝑦 → (𝜓𝜏))
1514elrab 3357 . . . . 5 (𝑦 ∈ {𝑧𝐵𝜓} ↔ (𝑦𝐵𝜏))
1613, 15anbi12i 732 . . . 4 ((𝑥 ∈ {𝑧𝐵𝜓} ∧ 𝑦 ∈ {𝑧𝐵𝜓}) ↔ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏)))
173adantr 481 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → 𝑀 ∈ Mnd)
18 simprll 801 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → 𝑥𝐵)
19 simprrl 803 . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → 𝑦𝐵)
20 issubmd.p . . . . . . 7 + = (+g𝑀)
214, 20mndcl 17282 . . . . . 6 ((𝑀 ∈ Mnd ∧ 𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
2217, 18, 19, 21syl3anc 1324 . . . . 5 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → (𝑥 + 𝑦) ∈ 𝐵)
23 an4 864 . . . . . 6 (((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏)) ↔ ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏)))
24 issubmd.cp . . . . . 6 ((𝜑 ∧ ((𝑥𝐵𝑦𝐵) ∧ (𝜃𝜏))) → 𝜂)
2523, 24sylan2b 492 . . . . 5 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → 𝜂)
26 issubmd.et . . . . . 6 (𝑧 = (𝑥 + 𝑦) → (𝜓𝜂))
2726elrab 3357 . . . . 5 ((𝑥 + 𝑦) ∈ {𝑧𝐵𝜓} ↔ ((𝑥 + 𝑦) ∈ 𝐵𝜂))
2822, 25, 27sylanbrc 697 . . . 4 ((𝜑 ∧ ((𝑥𝐵𝜃) ∧ (𝑦𝐵𝜏))) → (𝑥 + 𝑦) ∈ {𝑧𝐵𝜓})
2916, 28sylan2b 492 . . 3 ((𝜑 ∧ (𝑥 ∈ {𝑧𝐵𝜓} ∧ 𝑦 ∈ {𝑧𝐵𝜓})) → (𝑥 + 𝑦) ∈ {𝑧𝐵𝜓})
3029ralrimivva 2968 . 2 (𝜑 → ∀𝑥 ∈ {𝑧𝐵𝜓}∀𝑦 ∈ {𝑧𝐵𝜓} (𝑥 + 𝑦) ∈ {𝑧𝐵𝜓})
314, 5, 20issubm 17328 . . 3 (𝑀 ∈ Mnd → ({𝑧𝐵𝜓} ∈ (SubMnd‘𝑀) ↔ ({𝑧𝐵𝜓} ⊆ 𝐵0 ∈ {𝑧𝐵𝜓} ∧ ∀𝑥 ∈ {𝑧𝐵𝜓}∀𝑦 ∈ {𝑧𝐵𝜓} (𝑥 + 𝑦) ∈ {𝑧𝐵𝜓})))
323, 31syl 17 . 2 (𝜑 → ({𝑧𝐵𝜓} ∈ (SubMnd‘𝑀) ↔ ({𝑧𝐵𝜓} ⊆ 𝐵0 ∈ {𝑧𝐵𝜓} ∧ ∀𝑥 ∈ {𝑧𝐵𝜓}∀𝑦 ∈ {𝑧𝐵𝜓} (𝑥 + 𝑦) ∈ {𝑧𝐵𝜓})))
332, 11, 30, 32mpbir3and 1243 1 (𝜑 → {𝑧𝐵𝜓} ∈ (SubMnd‘𝑀))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1481  wcel 1988  wral 2909  {crab 2913  wss 3567  cfv 5876  (class class class)co 6635  Basecbs 15838  +gcplusg 15922  0gc0g 16081  Mndcmnd 17275  SubMndcsubmnd 17315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-riota 6596  df-ov 6638  df-0g 16083  df-mgm 17223  df-sgrp 17265  df-mnd 17276  df-submnd 17317
This theorem is referenced by:  mrcmndind  17347
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