Theorem issubmd 17349

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 3687	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵)
3		issubmd.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd)
4		issubmd.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑀)
5		issubmd.z	. . . . 5 ⊢ 0 = (0g‘𝑀)
6	4, 5	mndidcl 17308	. . . 4 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵)
7	3, 6	syl 17	. . 3 ⊢ (𝜑 → 0 ∈ 𝐵)
8		issubmd.cz	. . 3 ⊢ (𝜑 → 𝜒)
9		issubmd.ch	. . . 4 ⊢ (𝑧 = 0 → (𝜓 ↔ 𝜒))
10	9	elrab 3363	. . 3 ⊢ ( 0 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ ( 0 ∈ 𝐵 ∧ 𝜒))
11	7, 8, 10	sylanbrc 698	. 2 ⊢ (𝜑 → 0 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
12		issubmd.th	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃))
13	12	elrab 3363	. . . . 5 ⊢ (𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ (𝑥 ∈ 𝐵 ∧ 𝜃))
14		issubmd.ta	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏))
15	14	elrab 3363	. . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ (𝑦 ∈ 𝐵 ∧ 𝜏))
16	13, 15	anbi12i 733	. . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)))
17	3	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑀 ∈ Mnd)
18		simprll 802	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑥 ∈ 𝐵)
19		simprrl 804	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑦 ∈ 𝐵)
20		issubmd.p	. . . . . . 7 ⊢ + = (+g‘𝑀)
21	4, 20	mndcl 17301	. . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
22	17, 18, 19, 21	syl3anc 1326	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → (𝑥 + 𝑦) ∈ 𝐵)
23		an4 865	. . . . . 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 492	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝜂)
26		issubmd.et	. . . . . 6 ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂))
27	26	elrab 3363	. . . . 5 ⊢ ((𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ ((𝑥 + 𝑦) ∈ 𝐵 ∧ 𝜂))
28	22, 25, 27	sylanbrc 698	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
29	16, 28	sylan2b 492	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})) → (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
30	29	ralrimivva 2971	. 2 ⊢ (𝜑 → ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
31	4, 5, 20	issubm 17347	. . 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})))
32	3, 31	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})))
33	2, 11, 30, 32	mpbir3and 1245	1 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMnd‘𝑀))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ⊆ wss 3574  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  0_gc0g 16100  Mndcmnd 17294  SubMndcsubmnd 17334

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336

This theorem is referenced by:  mrcmndind  17366