Theorem issubm 18864

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 6885	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
2	1	pweqd 4584	. . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3		fveq2 6885	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (0g‘𝑚) = (0g‘𝑀))
4	3	eleq1d 2855	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((0g‘𝑚) ∈ 𝑡 ↔ (0g‘𝑀) ∈ 𝑡))
5		fveq2 6885	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
6	5	oveqd 7431	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥(+g‘𝑀)𝑦))
7	6	eleq1d 2855	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑡))
8	7	2ralbidv 3236	. . . . . 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 3441	. . . 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 18845	. . . 4 ⊢ SubMnd = (𝑚 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ((0g‘𝑚) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡)})
12		fvex 6898	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
13	12	pwex 5355	. . . . 5 ⊢ 𝒫 (Base‘𝑀) ∈ V
14	13	rabex 5313	. . . 4 ⊢ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)} ∈ V
15	10, 11, 14	fvmpt 6993	. . 3 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)})
16	15	eleq2d 2856	. 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 2859	. . . . 5 ⊢ (𝑡 = 𝑆 → ((0g‘𝑀) ∈ 𝑡 ↔ (0g‘𝑀) ∈ 𝑆))
18		eleq2 2859	. . . . . . 7 ⊢ (𝑡 = 𝑆 → ((𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
19	18	raleqbi1dv 3340	. . . . . 6 ⊢ (𝑡 = 𝑆 → (∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
20	19	raleqbi1dv 3340	. . . . 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 3658	. . 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 3974	. . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (Base‘𝑀))
25		issubm.z	. . . . . . 7 ⊢ 0 = (0g‘𝑀)
26	25	eleq1i 2861	. . . . . 6 ⊢ ( 0 ∈ 𝑆 ↔ (0g‘𝑀) ∈ 𝑆)
27		issubm.p	. . . . . . . . 9 ⊢ + = (+g‘𝑀)
28	27	oveqi 7427	. . . . . . . 8 ⊢ (𝑥 + 𝑦) = (𝑥(+g‘𝑀)𝑦)
29	28	eleq1i 2861	. . . . . . 7 ⊢ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
30	29	2ralbii 3147	. . . . . 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 1109	. . . 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 5308	. . . . 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)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2150  ∀wral 3086  {crab 3423   ⊆ wss 3913  𝒫 cpw 4567  ‘cfv 6540  (class class class)co 7414  Basecbs 17272  +_gcplusg 17313  0_gc0g 17495  Mndcmnd 18795  SubMndcsubmnd 18843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742  ax-sep 5262  ax-nul 5274  ax-pow 5340  ax-pr 5408

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ne 2966  df-ral 3087  df-rex 3097  df-rab 3424  df-v 3464  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5560  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-iota 6496  df-fun 6542  df-fv 6548  df-ov 7417  df-submnd 18845

This theorem is referenced by:  issubm2  18865  issubmd  18867  mndissubm  18868  submcl  18873  0subm  18879  insubm  18880  mhmima  18887  mhmeql  18888  submacs  18889  gsumwspan  18908  frmdsssubm  18923  sursubmefmnd  18958  injsubmefmnd  18959  issubg3  19214  cycsubm  19276  cntzsubm  19411  oppgsubm  19435  lsmsubm  19726  issubrg3  20688  isdomn3  20802  prmidlsubm  21470  cnsubmlem  21548  nn0srg  21570  rge0srg  21571  xrge0subm  21576  efsubm  26696  fxpsubm  33462  rrgsubm  33574  1arithufdlem4  33807  iistmd  34262  mon1psubm  43878