Theorem gexval 18755

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 18709	. . 3 ⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
3		nnex 11665	. . . . . 6 ⊢ ℕ ∈ V
4	3	rabex 5195	. . . . 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 6655	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
8		gexval.1	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	eqtr4di 2812	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
10	6	fveq2d 6655	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (.g‘𝑔) = (.g‘𝐺))
11		gexval.2	. . . . . . . . . . . . . 14 ⊢ · = (.g‘𝐺)
12	10, 11	eqtr4di 2812	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (.g‘𝑔) = · )
13	12	oveqd 7160	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (𝑦(.g‘𝑔)𝑥) = (𝑦 · 𝑥))
14	6	fveq2d 6655	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (0g‘𝑔) = (0g‘𝐺))
15		gexval.3	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝐺)
16	14, 15	eqtr4di 2812	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (0g‘𝑔) = 0 )
17	13, 16	eqeq12d 2775	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → ((𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 · 𝑥) = 0 ))
18	9, 17	raleqbidv 3317	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 ))
19	18	rabbidv 3390	. . . . . . . . 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 2812	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} = 𝐼)
22	21	eqeq2d 2770	. . . . . . 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 2761	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)}) → (𝑖 = ∅ ↔ 𝐼 = ∅))
25	23	infeq1d 8959	. . . . 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 4439	. . . 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 3837	. . 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 3427	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
29		c0ex 10658	. . . . 5 ⊢ 0 ∈ V
30		ltso 10744	. . . . . 6 ⊢ < Or ℝ
31	30	infex 8975	. . . . 5 ⊢ inf(𝐼, ℝ, < ) ∈ V
32	29, 31	ifex 4463	. . . 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 6760	. 2 ⊢ (𝐺 ∈ 𝑉 → (gEx‘𝐺) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
35	1, 34	syl5eq 2806	1 ⊢ (𝐺 ∈ 𝑉 → 𝐸 = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3068  {crab 3072  Vcvv 3407  ⦋csb 3801  ∅c0 4221  ifcif 4413  ‘cfv 6328  (class class class)co 7143  infcinf 8923  ℝcr 10559  0cc0 10560   < clt 10698  ℕcn 11659  Basecbs 16526  0_gc0g 16756  ._gcmg 18276  gExcgex 18705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5162  ax-nul 5169  ax-pow 5227  ax-pr 5291  ax-un 7452  ax-cnex 10616  ax-resscn 10617  ax-1cn 10618  ax-icn 10619  ax-addcl 10620  ax-mulcl 10622  ax-i2m1 10628  ax-pre-lttri 10634  ax-pre-lttrn 10635

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2899  df-ne 2950  df-nel 3054  df-ral 3073  df-rex 3074  df-reu 3075  df-rmo 3076  df-rab 3077  df-v 3409  df-sbc 3694  df-csb 3802  df-dif 3857  df-un 3859  df-in 3861  df-ss 3871  df-pss 3873  df-nul 4222  df-if 4414  df-pw 4489  df-sn 4516  df-pr 4518  df-tp 4520  df-op 4522  df-uni 4792  df-iun 4878  df-br 5026  df-opab 5088  df-mpt 5106  df-tr 5132  df-id 5423  df-eprel 5428  df-po 5436  df-so 5437  df-fr 5476  df-we 5478  df-xp 5523  df-rel 5524  df-cnv 5525  df-co 5526  df-dm 5527  df-rn 5528  df-res 5529  df-ima 5530  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7146  df-om 7573  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-er 8292  df-en 8521  df-dom 8522  df-sdom 8523  df-sup 8924  df-inf 8925  df-pnf 10700  df-mnf 10701  df-ltxr 10703  df-nn 11660  df-gex 18709

This theorem is referenced by:  gexlem1  18756  gexlem2  18759