Theorem gexval 19754

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 19705	. . 3 ⊢ gEx = (𝑔 ∈ V ↦ ⦋{𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} / 𝑖⦌if(𝑖 = ∅, 0, inf(𝑖, ℝ, < )))
3		nnex 12311	. . . . . 6 ⊢ ℕ ∈ V
4	3	rabex 5299	. . . . 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 6877	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = (Base‘𝐺))
8		gexval.1	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐺)
9	7, 8	eqtr4di 2813	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (Base‘𝑔) = 𝑋)
10	6	fveq2d 6877	. . . . . . . . . . . . . 14 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (.g‘𝑔) = (.g‘𝐺))
11		gexval.2	. . . . . . . . . . . . . 14 ⊢ · = (.g‘𝐺)
12	10, 11	eqtr4di 2813	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (.g‘𝑔) = · )
13	12	oveqd 7425	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (𝑦(.g‘𝑔)𝑥) = (𝑦 · 𝑥))
14	6	fveq2d 6877	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (0g‘𝑔) = (0g‘𝐺))
15		gexval.3	. . . . . . . . . . . . 13 ⊢ 0 = (0g‘𝐺)
16	14, 15	eqtr4di 2813	. . . . . . . . . . . 12 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (0g‘𝑔) = 0 )
17	13, 16	eqeq12d 2776	. . . . . . . . . . 11 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → ((𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ (𝑦 · 𝑥) = 0 ))
18	9, 17	raleqbidv 3334	. . . . . . . . . 10 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → (∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔) ↔ ∀𝑥 ∈ 𝑋 (𝑦 · 𝑥) = 0 ))
19	18	rabbidv 3419	. . . . . . . . 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 2813	. . . . . . . 8 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) → {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)} = 𝐼)
22	21	eqeq2d 2771	. . . . . . 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 2762	. . . . 5 ⊢ (((𝐺 ∈ 𝑉 ∧ 𝑔 = 𝐺) ∧ 𝑖 = {𝑦 ∈ ℕ ∣ ∀𝑥 ∈ (Base‘𝑔)(𝑦(.g‘𝑔)𝑥) = (0g‘𝑔)}) → (𝑖 = ∅ ↔ 𝐼 = ∅))
25	23	infeq1d 9448	. . . . 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 4508	. . . 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 3882	. . 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 3471	. . 3 ⊢ (𝐺 ∈ 𝑉 → 𝐺 ∈ V)
29		c0ex 11272	. . . . 5 ⊢ 0 ∈ V
30		ltso 11362	. . . . . 6 ⊢ < Or ℝ
31	30	infex 9465	. . . . 5 ⊢ inf(𝐼, ℝ, < ) ∈ V
32	29, 31	ifex 4532	. . . 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 6990	. 2 ⊢ (𝐺 ∈ 𝑉 → (gEx‘𝐺) = if(𝐼 = ∅, 0, inf(𝐼, ℝ, < )))
35	1, 34	eqtrid 2807	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 3076  {crab 3412  Vcvv 3450  ⦋csb 3846  ∅c0 4278  ifcif 4481  ‘cfv 6527  (class class class)co 7408  infcinf 9411  ℝcr 11171  0cc0 11172   < clt 11315  ℕcn 12305  Basecbs 17349  0_gc0g 17572  ._gcmg 19239  gExcgex 19701

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 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11228  ax-resscn 11229  ax-1cn 11230  ax-icn 11231  ax-addcl 11232  ax-mulcl 11234  ax-i2m1 11240  ax-pre-lttri 11246  ax-pre-lttrn 11247

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-ov 7411  df-om 7861  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8695  df-en 8952  df-dom 8953  df-sdom 8954  df-sup 9412  df-inf 9413  df-pnf 11317  df-mnf 11318  df-ltxr 11320  df-nn 12306  df-gex 19705

This theorem is used by:  gexlem1  19755  gexlem2  19758