Theorem issubm 18867

Description: Expand definition of a submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)

Hypotheses

Ref	Expression
issubm.b	⊢ 𝐵 = (Base‘𝑀)
issubm.z	⊢ 0 = (0g‘𝑀)
issubm.p	⊢ + = (+g‘𝑀)

Assertion

Ref	Expression
issubm	⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))

Distinct variable groups:   𝑥,𝑀,𝑦   𝑥,𝑆,𝑦

Allowed substitution hints:   𝐵(𝑥, 𝑦)   + (𝑥, 𝑦)   0 (𝑥, 𝑦)

Proof of Theorem issubm

Dummy variables 𝑚 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6881	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
2	1	pweqd 4578	. . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3		fveq2 6881	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (0g‘𝑚) = (0g‘𝑀))
4	3	eleq1d 2847	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((0g‘𝑚) ∈ 𝑡 ↔ (0g‘𝑀) ∈ 𝑡))
5		fveq2 6881	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
6	5	oveqd 7429	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥(+g‘𝑀)𝑦))
7	6	eleq1d 2847	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑡))
8	7	2ralbidv 3228	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡))
9	4, 8	anbi12d 643	. . . . 5 ⊢ (𝑚 = 𝑀 → (((0g‘𝑚) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡) ↔ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)))
10	2, 9	rabeqbidv 3433	. . . 4 ⊢ (𝑚 = 𝑀 → {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ((0g‘𝑚) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡)} = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)})
11		df-submnd 18848	. . . 4 ⊢ SubMnd = (𝑚 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ((0g‘𝑚) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡)})
12		fvex 6894	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
13	12	pwex 5350	. . . . 5 ⊢ 𝒫 (Base‘𝑀) ∈ V
14	13	rabex 5308	. . . 4 ⊢ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)} ∈ V
15	10, 11, 14	fvmpt 6989	. . 3 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)})
16	15	eleq2d 2848	. 2 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)}))
17		eleq2 2851	. . . . 5 ⊢ (𝑡 = 𝑆 → ((0g‘𝑀) ∈ 𝑡 ↔ (0g‘𝑀) ∈ 𝑆))
18		eleq2 2851	. . . . . . 7 ⊢ (𝑡 = 𝑆 → ((𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
19	18	raleqbi1dv 3332	. . . . . 6 ⊢ (𝑡 = 𝑆 → (∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
20	19	raleqbi1dv 3332	. . . . 5 ⊢ (𝑡 = 𝑆 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
21	17, 20	anbi12d 643	. . . 4 ⊢ (𝑡 = 𝑆 → (((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡) ↔ ((0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)))
22	21	elrab 3649	. . 3 ⊢ (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)} ↔ (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ((0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)))
23		issubm.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑀)
24	23	sseq2i 3965	. . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (Base‘𝑀))
25		issubm.z	. . . . . . 7 ⊢ 0 = (0g‘𝑀)
26	25	eleq1i 2853	. . . . . 6 ⊢ ( 0 ∈ 𝑆 ↔ (0g‘𝑀) ∈ 𝑆)
27		issubm.p	. . . . . . . . 9 ⊢ + = (+g‘𝑀)
28	27	oveqi 7425	. . . . . . . 8 ⊢ (𝑥 + 𝑦) = (𝑥(+g‘𝑀)𝑦)
29	28	eleq1i 2853	. . . . . . 7 ⊢ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
30	29	2ralbii 3139	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
31	26, 30	anbi12i 639	. . . . 5 ⊢ (( 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ ((0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
32	24, 31	anbi12i 639	. . . 4 ⊢ ((𝑆 ⊆ 𝐵 ∧ ( 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ((0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)))
33		3anass 1110	. . . 4 ⊢ ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ 𝐵 ∧ ( 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
34	12	elpw2 5304	. . . . 5 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
35	34	anbi1i 635	. . . 4 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ((0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)) ↔ (𝑆 ⊆ (Base‘𝑀) ∧ ((0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)))
36	32, 33, 35	3bitr4ri 307	. . 3 ⊢ ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ((0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))
37	22, 36	bitri 278	. 2 ⊢ (𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)} ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))
38	16, 37	bitrdi 290	1 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  ∀wral 3078  {crab 3415   ⊆ wss 3904  𝒫 cpw 4561  ‘cfv 6536  (class class class)co 7412  Basecbs 17275  +_gcplusg 17316  0_gc0g 17498  Mndcmnd 18798  SubMndcsubmnd 18846

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5555  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7415  df-submnd 18848

This theorem is used by:  issubm2  18868  issubmd  18870  mndissubm  18871  submcl  18876  0subm  18882  insubm  18883  mhmima  18890  mhmeql  18891  submacs  18892  gsumwspan  18911  frmdsssubm  18926  sursubmefmnd  18961  injsubmefmnd  18962  issubg3  19217  cycsubm  19279  cntzsubm  19414  oppgsubm  19438  lsmsubm  19729  issubrg3  20710  isdomn3  20824  prmidlsubm  21498  cnsubmlem  21576  nn0srg  21598  rge0srg  21599  xrge0subm  21604  efsubm  26727  fxpsubm  33501  rrgsubm  33613  1arithufdlem4  33846  iistmd  34301  mon1psubm  43954