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	⊢ + = (+g‘𝑀)
rabsubmgmd.m	⊢ (𝜑 → 𝑀 ∈ Mgm)
rabsubmgmd.cp	⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂)
rabsubmgmd.th	⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃))
rabsubmgmd.ta	⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏))
rabsubmgmd.et	⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂))

Assertion

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

Step	Hyp	Ref	Expression
1		ssrab2 3650	. . 3 ⊢ {𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵
2	1	a1i 11	. 2 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵)
3		rabsubmgmd.th	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃))
4	3	elrab 3331	. . . . 5 ⊢ (𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ (𝑥 ∈ 𝐵 ∧ 𝜃))
5		rabsubmgmd.ta	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏))
6	5	elrab 3331	. . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ (𝑦 ∈ 𝐵 ∧ 𝜏))
7	4, 6	anbi12i 729	. . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)))
8		rabsubmgmd.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ Mgm)
9	8	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑀 ∈ Mgm)
10		simprll 798	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑥 ∈ 𝐵)
11		simprrl 800	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑦 ∈ 𝐵)
12		rabsubmgmd.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
13		rabsubmgmd.p	. . . . . . 7 ⊢ + = (+g‘𝑀)
14	12, 13	mgmcl 17068	. . . . . 6 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
15	9, 10, 11, 14	syl3anc 1318	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → (𝑥 + 𝑦) ∈ 𝐵)
16		simpl 472	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝜃) → 𝑥 ∈ 𝐵)
17		simpl 472	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜏) → 𝑦 ∈ 𝐵)
18	16, 17	anim12i 588	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
19		simpr 476	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝜃) → 𝜃)
20		simpr 476	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜏) → 𝜏)
21	19, 20	anim12i 588	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)) → (𝜃 ∧ 𝜏))
22	18, 21	jca 553	. . . . . 6 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏)))
23		rabsubmgmd.cp	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂)
24	22, 23	sylan2 490	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝜂)
25		rabsubmgmd.et	. . . . . 6 ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂))
26	25	elrab 3331	. . . . 5 ⊢ ((𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ ((𝑥 + 𝑦) ∈ 𝐵 ∧ 𝜂))
27	15, 24, 26	sylanbrc 695	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
28	7, 27	sylan2b 491	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})) → (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
29	28	ralrimivva 2954	. 2 ⊢ (𝜑 → ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
30	12, 13	issubmgm 41579	. . 3 ⊢ (𝑀 ∈ Mgm → ({𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMgm‘𝑀) ↔ ({𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵 ∧ ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})))
31	8, 30	syl 17	. 2 ⊢ (𝜑 → ({𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMgm‘𝑀) ↔ ({𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵 ∧ ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})))
32	2, 29, 31	mpbir2and 959	1 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMgm‘𝑀))

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)