Theorem gexval 19647

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 19598	. . 3 ⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
3		nnex 12238	. . . . . 6 ⊢ ℕ ∈ V
4	3	rabex 5309	. . . . 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 489	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → 𝑔 = 𝐺)
7	6	fveq2d 6885	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
8		gexval.1	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	eqtr4di 2814	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
10	6	fveq2d 6885	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (.g‘𝑔) = (.g‘𝐺))
11		gexval.2	. . . . . . . . . . . . . 14 ⊢ · = (.g‘𝐺)
12	10, 11	eqtr4di 2814	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (.g‘𝑔) = · )
13	12	oveqd 7427	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (𝑦(.g‘𝑔)𝑥) = (𝑦 · 𝑥))
14	6	fveq2d 6885	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (0g‘𝑔) = (0g‘𝐺))
15		gexval.3	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝐺)
16	14, 15	eqtr4di 2814	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (0g‘𝑔) = 0 )
17	13, 16	eqeq12d 2777	. . . . . . . . . . 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 2814	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} = 𝐼)
22	21	eqeq2d 2772	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} ↔ 𝑖 = 𝐼))
23	22	biimpa 481	. . . . . 6 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)}) → 𝑖 = 𝐼)
24	23	eqeq1d 2763	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)}) → (𝑖 = ∅ ↔ 𝐼 = ∅))
25	23	infeq1d 9437	. . . . 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 4513	. . . 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 3888	. . 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 11199	. . . . 5 ⊢ 0 ∈ V
30		ltso 11289	. . . . . 6 ⊢ < Or ℝ
31	30	infex 9454	. . . . 5 ⊢ inf(𝐼, ℝ, < ) ∈ V
32	29, 31	ifex 4537	. . . 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 6998	. 2 ⊢ (𝐺 ∈ 𝑉 → (gEx‘𝐺) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
35	1, 34	eqtrid 2808	1 ⊢ (𝐺 ∈ 𝑉 → 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568   ∈ wcel 2141  ∀wral 3077  {crab 3414  Vcvv 3453  ⦋csb 3852  ∅c0 4285  ifcif 4486  ‘cfv 6536  (class class class)co 7410  infcinf 9400  ℝcr 11098  0cc0 11099   < clt 11242  ℕcn 12232  Basecbs 17268  0_gc0g 17491  ._gcmg 19132  gExcgex 19594

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 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11155  ax-resscn 11156  ax-1cn 11157  ax-icn 11158  ax-addcl 11159  ax-mulcl 11161  ax-i2m1 11167  ax-pre-lttri 11173  ax-pre-lttrn 11174

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7862  df-2nd 7986  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8693  df-en 8943  df-dom 8944  df-sdom 8945  df-sup 9401  df-inf 9402  df-pnf 11244  df-mnf 11245  df-ltxr 11247  df-nn 12233  df-gex 19598

This theorem is referenced by:  gexlem1  19648  gexlem2  19651