Theorem gexval 19709

Description: Value of the exponent of a group. (Contributed by Mario Carneiro, 23-Apr-2016.) (Revised by AV, 26-Sep-2020.)

Hypotheses

Ref	Expression
gexval.1	⊢ 𝑋 = (Base‘𝐺)
gexval.2	⊢ · = (.g‘𝐺)
gexval.3	⊢ 0 = (0g‘𝐺)
gexval.4	⊢ 𝐸 = (gEx‘𝐺)
gexval.i	⊢ 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }

Assertion

Ref	Expression
gexval	⊢ (𝐺 ∈ 𝑉 → 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))

Distinct variable groups:   𝑥,𝑦, 0   𝑥,𝐺,𝑦   𝑥,𝑉,𝑦   𝑥, · ,𝑦   𝑥,𝑋

Allowed substitution hints:   𝐸(𝑥, 𝑦)   𝐼(𝑥, 𝑦)   𝑋(𝑦)

Proof of Theorem gexval

Dummy variables 𝑔 𝑖 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gexval.4	. 2 ⊢ 𝐸 = (gEx‘𝐺)
2		df-gex 19660	. . 3 ⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
3		nnex 12266	. . . . . 6 ⊢ ℕ ∈ V
4	3	rabex 5307	. . . . 5 ⊢ {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} ∈ V
5	4	a1i 11	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} ∈ V)
6		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
7	6	fveq2d 6886	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
8		gexval.1	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	eqtr4di 2815	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
10	6	fveq2d 6886	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (.g‘𝑔) = (.g‘𝐺))
11		gexval.2	. . . . . . . . . . . . . 14 ⊢ · = (.g‘𝐺)
12	10, 11	eqtr4di 2815	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (.g‘𝑔) = · )
13	12	oveqd 7433	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (𝑦(.g‘𝑔)𝑥) = (𝑦 · 𝑥))
14	6	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (0g‘𝑔) = (0g‘𝐺))
15		gexval.3	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝐺)
16	14, 15	eqtr4di 2815	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (0g‘𝑔) = 0 )
17	13, 16	eqeq12d 2778	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → ((𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 · 𝑥) = 0 ))
18	9, 17	raleqbidv 3336	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 ))
19	18	rabbidv 3421	. . . . . . . . 9 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 })
20		gexval.i	. . . . . . . . 9 ⊢ 𝐼 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 }
21	19, 20	eqtr4di 2815	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} = 𝐼)
22	21	eqeq2d 2773	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} ↔ 𝑖 = 𝐼))
23	22	biimpa 482	. . . . . 6 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)}) → 𝑖 = 𝐼)
24	23	eqeq1d 2764	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)}) → (𝑖 = ∅ ↔ 𝐼 = ∅))
25	23	infeq1d 9451	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)}) → inf(𝑖, ℝ, < ) = inf(𝐼, ℝ, < ))
26	24, 25	ifbieq2d 4512	. . . 4 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)}) → if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
27	5, 26	csbied 3886	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → ⦋{𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
28		elex 3474	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
29		c0ex 11227	. . . . 5 ⊢ 0 ∈ V
30		ltso 11317	. . . . . 6 ⊢ < Or ℝ
31	30	infex 9468	. . . . 5 ⊢ inf(𝐼, ℝ, < ) ∈ V
32	29, 31	ifex 4536	. . . 4 ⊢ if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V
33	32	a1i 11	. . 3 ⊢ (𝐺 ∈ 𝑉 → if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )) ∈ V)
34	2, 27, 28, 33	fvmptd2 6999	. 2 ⊢ (𝐺 ∈ 𝑉 → (gEx‘𝐺) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
35	1, 34	eqtrid 2809	1 ⊢ (𝐺 ∈ 𝑉 → 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  {crab 3414  Vcvv 3453  ⦋csb 3850  ∅c0 4282  ifcif 4485  ‘cfv 6537  (class class class)co 7416  infcinf 9414  ℝcr 11126  0cc0 11127   < clt 11270  ℕcn 12260  Basecbs 17305  0_gc0g 17528  ._gcmg 19194  gExcgex 19656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-cnex 11183  ax-resscn 11184  ax-1cn 11185  ax-icn 11186  ax-addcl 11187  ax-mulcl 11189  ax-i2m1 11195  ax-pre-lttri 11201  ax-pre-lttrn 11202

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7419  df-om 7866  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-er 8699  df-en 8956  df-dom 8957  df-sdom 8958  df-sup 9415  df-inf 9416  df-pnf 11272  df-mnf 11273  df-ltxr 11275  df-nn 12261  df-gex 19660

This theorem is used by:  gexlem1  19710  gexlem2  19713