Theorem issubm 17347

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 6191	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
2	1	pweqd 4163	. . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
3		fveq2 6191	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (0g‘𝑚) = (0g‘𝑀))
4	3	eleq1d 2686	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((0g‘𝑚) ∈ 𝑡 ↔ (0g‘𝑀) ∈ 𝑡))
5		fveq2 6191	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
6	5	oveqd 6667	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥(+g‘𝑀)𝑦))
7	6	eleq1d 2686	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑡))
8	7	2ralbidv 2989	. . . . . 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 747	. . . . 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 3195	. . . 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 17336	. . . 4 ⊢ SubMnd = (𝑚 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ((0g‘𝑚) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡)})
12		fvex 6201	. . . . . 6 ⊢ (Base‘𝑀) ∈ V
13	12	pwex 4848	. . . . 5 ⊢ 𝒫 (Base‘𝑀) ∈ V
14	13	rabex 4813	. . . 4 ⊢ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)} ∈ V
15	10, 11, 14	fvmpt 6282	. . 3 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)})
16	15	eleq2d 2687	. 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 2690	. . . . 5 ⊢ (𝑡 = 𝑆 → ((0g‘𝑀) ∈ 𝑡 ↔ (0g‘𝑀) ∈ 𝑆))
18		eleq2 2690	. . . . . . 7 ⊢ (𝑡 = 𝑆 → ((𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
19	18	raleqbi1dv 3146	. . . . . 6 ⊢ (𝑡 = 𝑆 → (∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
20	19	raleqbi1dv 3146	. . . . 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 747	. . . 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 3363	. . 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 3630	. . . . 5 ⊢ (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (Base‘𝑀))
25		issubm.z	. . . . . . 7 ⊢ 0 = (0g‘𝑀)
26	25	eleq1i 2692	. . . . . 6 ⊢ ( 0 ∈ 𝑆 ↔ (0g‘𝑀) ∈ 𝑆)
27		issubm.p	. . . . . . . . 9 ⊢ + = (+g‘𝑀)
28	27	oveqi 6663	. . . . . . . 8 ⊢ (𝑥 + 𝑦) = (𝑥(+g‘𝑀)𝑦)
29	28	eleq1i 2692	. . . . . . 7 ⊢ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
30	29	2ralbii 2981	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
31	26, 30	anbi12i 733	. . . . 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 733	. . . 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 1042	. . . 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 4828	. . . . 5 ⊢ (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀))
35	34	anbi1i 731	. . . 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 293	. . 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 264	. 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	syl6bb 276	1 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916   ⊆ wss 3574  𝒫 cpw 4158  ‘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-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-ral 2917  df-rex 2918  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-ov 6653  df-submnd 17336

This theorem is referenced by:  issubm2  17348  issubmd  17349  submcl  17353  mhmima  17363  mhmeql  17364  submacs  17365  gsumwspan  17383  frmdsssubm  17398  issubg3  17612  cntzsubm  17768  oppgsubm  17792  lsmsubm  18068  issubrg3  18808  xrge0subm  19787  cnsubmlem  19794  nn0srg  19816  rge0srg  19817  efsubm  24297  iistmd  29948  isdomn3  37782  mon1psubm  37784