Theorem issubm 13756

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		df-submnd 13744	. . . 4 ⊢ SubMnd = (𝑚 ∈ Mnd ↦ {𝑡 ∈ 𝒫 (Base‘𝑚) ∣ ((0g‘𝑚) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡)})
2		fveq2 5690	. . . . . 6 ⊢ (𝑚 = 𝑀 → (Base‘𝑚) = (Base‘𝑀))
3	2	pweqd 3690	. . . . 5 ⊢ (𝑚 = 𝑀 → 𝒫 (Base‘𝑚) = 𝒫 (Base‘𝑀))
4		fveq2 5690	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (0g‘𝑚) = (0g‘𝑀))
5	4	eleq1d 2307	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((0g‘𝑚) ∈ 𝑡 ↔ (0g‘𝑀) ∈ 𝑡))
6		fveq2 5690	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (+g‘𝑚) = (+g‘𝑀))
7	6	oveqd 6092	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (𝑥(+g‘𝑚)𝑦) = (𝑥(+g‘𝑀)𝑦))
8	7	eleq1d 2307	. . . . . . 7 ⊢ (𝑚 = 𝑀 → ((𝑥(+g‘𝑚)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑡))
9	8	2ralbidv 2574	. . . . . 6 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑚)𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡))
10	5, 9	anbi12d 477	. . . . 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)))
11	3, 10	rabeqbidv 2816	. . . 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)})
12		id 19	. . . 4 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ Mnd)
13		basfn 13389	. . . . . . 7 ⊢ Base Fn V
14		elex 2833	. . . . . . 7 ⊢ (𝑀 ∈ Mnd → 𝑀 ∈ V)
15		funfvex 5707	. . . . . . . 8 ⊢ ((Fun Base ∧ 𝑀 ∈ dom Base) → (Base‘𝑀) ∈ V)
16	15	funfni 5478	. . . . . . 7 ⊢ ((Base Fn V ∧ 𝑀 ∈ V) → (Base‘𝑀) ∈ V)
17	13, 14, 16	sylancr 418	. . . . . 6 ⊢ (𝑀 ∈ Mnd → (Base‘𝑀) ∈ V)
18	17	pwexd 4313	. . . . 5 ⊢ (𝑀 ∈ Mnd → 𝒫 (Base‘𝑀) ∈ V)
19		rabexg 4274	. . . . 5 ⊢ (𝒫 (Base‘𝑀) ∈ V → {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)} ∈ V)
20	18, 19	syl 14	. . . 4 ⊢ (𝑀 ∈ Mnd → {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)} ∈ V)
21	1, 11, 12, 20	fvmptd3 5793	. . 3 ⊢ (𝑀 ∈ Mnd → (SubMnd‘𝑀) = {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)})
22	21	eleq2d 2308	. 2 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ 𝑆 ∈ {𝑡 ∈ 𝒫 (Base‘𝑀) ∣ ((0g‘𝑀) ∈ 𝑡 ∧ ∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡)}))
23		eleq2 2302	. . . . 5 ⊢ (𝑡 = 𝑆 → ((0g‘𝑀) ∈ 𝑡 ↔ (0g‘𝑀) ∈ 𝑆))
24		eleq2 2302	. . . . . . 7 ⊢ (𝑡 = 𝑆 → ((𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
25	24	raleqbi1dv 2761	. . . . . 6 ⊢ (𝑡 = 𝑆 → (∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
26	25	raleqbi1dv 2761	. . . . 5 ⊢ (𝑡 = 𝑆 → (∀𝑥 ∈ 𝑡 ∀𝑦 ∈ 𝑡 (𝑥(+g‘𝑀)𝑦) ∈ 𝑡 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
27	23, 26	anbi12d 477	. . . 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)))
28	27	elrab 2982	. . 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)))
29		issubm.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑀)
30	29	sseq2i 3275	. . . . . 6 ⊢ (𝑆 ⊆ 𝐵 ↔ 𝑆 ⊆ (Base‘𝑀))
31		issubm.z	. . . . . . . 8 ⊢ 0 = (0g‘𝑀)
32	31	eleq1i 2304	. . . . . . 7 ⊢ ( 0 ∈ 𝑆 ↔ (0g‘𝑀) ∈ 𝑆)
33		issubm.p	. . . . . . . . . 10 ⊢ + = (+g‘𝑀)
34	33	oveqi 6088	. . . . . . . . 9 ⊢ (𝑥 + 𝑦) = (𝑥(+g‘𝑀)𝑦)
35	34	eleq1i 2304	. . . . . . . 8 ⊢ ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
36	35	2ralbii 2558	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆 ↔ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆)
37	32, 36	anbi12i 464	. . . . . 6 ⊢ (( 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ ((0g‘𝑀) ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥(+g‘𝑀)𝑦) ∈ 𝑆))
38	30, 37	anbi12i 464	. . . . 5 ⊢ ((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)))
39	38	a1i 9	. . . 4 ⊢ (𝑀 ∈ Mnd → ((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))))
40		3anass 1013	. . . . 5 ⊢ ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ 𝐵 ∧ ( 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))
41	40	a1i 9	. . . 4 ⊢ (𝑀 ∈ Mnd → ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆) ↔ (𝑆 ⊆ 𝐵 ∧ ( 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆))))
42		elpw2g 4287	. . . . . 6 ⊢ ((Base‘𝑀) ∈ V → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
43	17, 42	syl 14	. . . . 5 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ 𝒫 (Base‘𝑀) ↔ 𝑆 ⊆ (Base‘𝑀)))
44	43	anbi1d 469	. . . 4 ⊢ (𝑀 ∈ Mnd → ((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))))
45	39, 41, 44	3bitr4rd 221	. . 3 ⊢ (𝑀 ∈ Mnd → ((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)))
46	28, 45	bitrid 192	. 2 ⊢ (𝑀 ∈ Mnd → (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)))
47	22, 46	bitrd 188	1 ⊢ (𝑀 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝑀) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ ∀𝑥 ∈ 𝑆 ∀𝑦 ∈ 𝑆 (𝑥 + 𝑦) ∈ 𝑆)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  {crab 2532  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3685   Fn wfn 5367  ‘cfv 5372  (class class class)co 6075  Basecbs 13330  +_gcplusg 13408  0_gc0g 13587  Mndcmnd 13706  SubMndcsubmnd 13742

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-fv 5380  df-ov 6078  df-inn 9284  df-ndx 13333  df-slot 13334  df-base 13336  df-submnd 13744

This theorem is referenced by:  issubm2  13757  issubmd  13758  mndissubm  13759  submss  13760  submid  13761  subm0cl  13762  submcl  13763  0subm  13768  insubm  13769  mhmima  13775  mhmeql  13776  issubg3  13972  issubrg3  14528  cnsubmlem  14887