Theorem rabsubmgmd 45233

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 4009	. . 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 3617	. . . . 5 ⊢ (𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ (𝑥 ∈ 𝐵 ∧ 𝜃))
5		rabsubmgmd.ta	. . . . . 6 ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏))
6	5	elrab 3617	. . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ (𝑦 ∈ 𝐵 ∧ 𝜏))
7	4, 6	anbi12i 626	. . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)))
8		rabsubmgmd.et	. . . . 5 ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂))
9		rabsubmgmd.m	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ Mgm)
10	9	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑀 ∈ Mgm)
11		simprll 775	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑥 ∈ 𝐵)
12		simprrl 777	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑦 ∈ 𝐵)
13		rabsubmgmd.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
14		rabsubmgmd.p	. . . . . . 7 ⊢ + = (+g‘𝑀)
15	13, 14	mgmcl 18244	. . . . . 6 ⊢ ((𝑀 ∈ Mgm ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵)
16	10, 11, 12, 15	syl3anc 1369	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → (𝑥 + 𝑦) ∈ 𝐵)
17		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝜃) → 𝑥 ∈ 𝐵)
18		simpl 482	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜏) → 𝑦 ∈ 𝐵)
19	17, 18	anim12i 612	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)) → (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵))
20		simpr 484	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝜃) → 𝜃)
21		simpr 484	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝜏) → 𝜏)
22	20, 21	anim12i 612	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)) → (𝜃 ∧ 𝜏))
23	19, 22	jca 511	. . . . . 6 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏)))
24		rabsubmgmd.cp	. . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂)
25	23, 24	sylan2 592	. . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝜂)
26	8, 16, 25	elrabd 3619	. . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
27	7, 26	sylan2b 593	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})) → (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
28	27	ralrimivva 3114	. 2 ⊢ (𝜑 → ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})
29	13, 14	issubmgm 45231	. . 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})))
30	9, 29	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})))
31	2, 28, 30	mpbir2and 709	1 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMgm‘𝑀))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067   ⊆ wss 3883  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  Mgmcmgm 18239  SubMgmcsubmgm 45220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-ov 7258  df-mgm 18241  df-submgm 45222

This theorem is referenced by: (None)