Theorem issubmd 18445

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	⊢ + = (+g‘𝑀)
issubmd.z	⊢ 0 = (0g‘𝑀)
issubmd.m	⊢ (𝜑 → 𝑀 ∈ Mnd)
issubmd.cz	⊢ (𝜑 → 𝜒)
issubmd.cp	⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂)
issubmd.ch	⊢ (𝑧 = 0 → (𝜓 ↔ 𝜒))
issubmd.th	⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃))
issubmd.ta	⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏))
issubmd.et	⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂))

Assertion

Ref	Expression
issubmd	⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (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

Step	Hyp	Ref	Expression
1		ssrab2 4013	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵)
3		issubmd.ch	. . 3 ⊢ (𝑧 = 0 → (𝜓 ↔ 𝜒))
4		issubmd.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd)
5		issubmd.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
6		issubmd.z	. . . . 5 ⊢ 0 = (0g‘𝑀)
7	5, 6	mndidcl 18400	. . . 4 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵)
8	4, 7	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ 𝐵)
9		issubmd.cz	. . 3 ⊢ (𝜑 → 𝜒)
10	3, 8, 9	elrabd 3626	. 2 ⊢ (𝜑 → 0 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
11		issubmd.th	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃))
12	11	elrab 3624	. . . . 5 ⊢ (𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ (𝑥 ∈ 𝐵 ∧ 𝜃))
13		issubmd.ta	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏))
14	13	elrab 3624	. . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ (𝑦 ∈ 𝐵 ∧ 𝜏))
15	12, 14	anbi12i 627	. . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)))
16		issubmd.et	. . . . 5 ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂))
17	4	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑀 ∈ Mnd)
18		simprll 776	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑥 ∈ 𝐵)
19		simprrl 778	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑦 ∈ 𝐵)
20		issubmd.p	. . . . . . 7 ⊢ + = (+g‘𝑀)
21	5, 20	mndcl 18393	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
22	17, 18, 19, 21	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → (𝑥 + 𝑦) ∈ 𝐵)
23		an4 653	. . . . . 6 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏)))
24		issubmd.cp	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂)
25	23, 24	sylan2b 594	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝜂)
26	16, 22, 25	elrabd 3626	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
27	15, 26	sylan2b 594	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})) → (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
28	27	ralrimivva 3123	. 2 ⊢ (𝜑 → ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
29	5, 6, 20	issubm 18442	. . 3 ⊢ (𝑀 ∈ Mnd → ({𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMnd‘𝑀) ↔ ({𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵 ∧ 0 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})))
30	4, 29	syl 17	. 2 ⊢ (𝜑 → ({𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMnd‘𝑀) ↔ ({𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵 ∧ 0 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})))
31	2, 10, 28, 30	mpbir3and 1341	1 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  {crab 3068   ⊆ wss 3887  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  0_gc0g 17150  Mndcmnd 18385  SubMndcsubmnd 18429

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-riota 7232  df-ov 7278  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431

This theorem is referenced by:  mndind  18466