Theorem sylow1 19730

Description: Sylow's first theorem. If 𝑃↑𝑁 is a prime power that divides the cardinality of 𝐺, then 𝐺 has a supgroup with size 𝑃↑𝑁. This is part of Metamath 100 proof #72. (Contributed by Mario Carneiro, 16-Jan-2015.)

Hypotheses

Ref	Expression
sylow1.x	⊢ 𝑋 = (Base‘𝐺)
sylow1.g	⊢ (𝜑 → 𝐺 ∈ Grp)
sylow1.f	⊢ (𝜑 → 𝑋 ∈ Fin)
sylow1.p	⊢ (𝜑 → 𝑃 ∈ ℙ)
sylow1.n	⊢ (𝜑 → 𝑁 ∈ ℕ0)
sylow1.d	⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))

Assertion

Ref	Expression
sylow1	⊢ (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃↑𝑁))

Distinct variable groups:   𝑔,𝑁   𝑔,𝑋   𝑔,𝐺   𝑃,𝑔   𝜑,𝑔

Proof of Theorem sylow1

Dummy variables 𝑎 𝑏 𝑠 𝑢 𝑥 𝑦 𝑧 ℎ 𝑘 𝑡 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sylow1.x	. . 3 ⊢ 𝑋 = (Base‘𝐺)
2		sylow1.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Grp)
3		sylow1.f	. . 3 ⊢ (𝜑 → 𝑋 ∈ Fin)
4		sylow1.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℙ)
5		sylow1.n	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
6		sylow1.d	. . 3 ⊢ (𝜑 → (𝑃↑𝑁) ∥ (♯‘𝑋))
7		eqid 2760	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
8		eqid 2760	. . 3 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}
9		oveq2 7421	. . . . . . 7 ⊢ (𝑠 = 𝑧 → (𝑢(+g‘𝐺)𝑠) = (𝑢(+g‘𝐺)𝑧))
10	9	cbvmptv 5209	. . . . . 6 ⊢ (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑧))
11		oveq1 7420	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑢(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)𝑧))
12	11	mpteq2dv 5199	. . . . . 6 ⊢ (𝑢 = 𝑥 → (𝑧 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑧)) = (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
13	10, 12	eqtrid 2807	. . . . 5 ⊢ (𝑢 = 𝑥 → (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
14	13	rneqd 5922	. . . 4 ⊢ (𝑢 = 𝑥 → ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = ran (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
15		mpteq1 5194	. . . . 5 ⊢ (𝑣 = 𝑦 → (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)) = (𝑧 ∈ 𝑦 ↦ (𝑥(+g‘𝐺)𝑧)))
16	15	rneqd 5922	. . . 4 ⊢ (𝑣 = 𝑦 → ran (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)) = ran (𝑧 ∈ 𝑦 ↦ (𝑥(+g‘𝐺)𝑧)))
17	14, 16	cbvmpov 7508	. . 3 ⊢ (𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠))) = (𝑥 ∈ 𝑋, 𝑦 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑧 ∈ 𝑦 ↦ (𝑥(+g‘𝐺)𝑧)))
18		preq12 4696	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → {𝑎, 𝑏} = {𝑥, 𝑦})
19	18	sseq1d 3962	. . . . 5 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↔ {𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}))
20		oveq2 7421	. . . . . . 7 ⊢ (𝑎 = 𝑥 → (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑥))
21		id 23	. . . . . . 7 ⊢ (𝑏 = 𝑦 → 𝑏 = 𝑦)
22	20, 21	eqeqan12d 2774	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ((𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏 ↔ (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑥) = 𝑦))
23	22	rexbidv 3186	. . . . 5 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏 ↔ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑥) = 𝑦))
24	19, 23	anbi12d 644	. . . 4 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → (({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏) ↔ ({𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑥) = 𝑦)))
25	24	cbvopabv 5178	. . 3 ⊢ {⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)} = {⟨𝑥, 𝑦⟩ ∣ ({𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑥) = 𝑦)}
26	1, 2, 3, 4, 5, 6, 7, 8, 17, 25	sylow1lem3 19727	. 2 ⊢ (𝜑 → ∃ℎ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} (𝑃 pCnt (♯‘[ℎ]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
27	2	adantr 486	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ (𝑃 pCnt (♯‘[ℎ]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → 𝐺 ∈ Grp)
28	3	adantr 486	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ (𝑃 pCnt (♯‘[ℎ]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → 𝑋 ∈ Fin)
29	4	adantr 486	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ (𝑃 pCnt (♯‘[ℎ]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → 𝑃 ∈ ℙ)
30	5	adantr 486	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ (𝑃 pCnt (♯‘[ℎ]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → 𝑁 ∈ ℕ0)
31	6	adantr 486	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ (𝑃 pCnt (♯‘[ℎ]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → (𝑃↑𝑁) ∥ (♯‘𝑋))
32		simprl 783	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ (𝑃 pCnt (♯‘[ℎ]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → ℎ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)})
33		eqid 2760	. . 3 ⊢ {𝑡 ∈ 𝑋 ∣ (𝑡(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))ℎ) = ℎ} = {𝑡 ∈ 𝑋 ∣ (𝑡(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))ℎ) = ℎ}
34		simprr 785	. . 3 ⊢ ((𝜑 ∧ (ℎ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ (𝑃 pCnt (♯‘[ℎ]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → (𝑃 pCnt (♯‘[ℎ]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))
35	1, 27, 28, 29, 30, 31, 7, 8, 17, 25, 32, 33, 34	sylow1lem5 19729	. 2 ⊢ ((𝜑 ∧ (ℎ ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ (𝑃 pCnt (♯‘[ℎ]{⟨𝑎, 𝑏⟩ ∣ ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ∧ ∃𝑘 ∈ 𝑋 (𝑘(𝑢 ∈ 𝑋, 𝑣 ∈ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↦ ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)))𝑎) = 𝑏)})) ≤ ((𝑃 pCnt (♯‘𝑋)) − 𝑁))) → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃↑𝑁))
36	26, 35	rexlimddv 3169	1 ⊢ (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃↑𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3086  {crab 3412   ⊆ wss 3899  𝒫 cpw 4557  {cpr 4586   class class class wbr 5103  {copab 5167   ↦ cmpt 5186  ran crn 5656  ‘cfv 6533  (class class class)co 7413   ∈ cmpo 7415  [cec 8694  Fincfn 8952   ≤ cle 11268   − cmin 11465  ℕ₀cn0 12528  ↑cexp 14125  ♯chash 14394   ∥ cdvds 16342  ℙcprime 16761   pCnt cpc 16928  Basecbs 17301  +_gcplusg 17342  Grpcgrp 19057  SubGrpcsubg 19243

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-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736  ax-inf2 9620  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 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 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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-disj 5071  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7863  df-1st 7986  df-2nd 7987  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-oadd 8459  df-er 8696  df-ec 8698  df-qs 8702  df-map 8828  df-en 8953  df-dom 8954  df-sdom 8955  df-fin 8956  df-sup 9412  df-inf 9413  df-oi 9482  df-dju 9906  df-card 9944  df-pnf 11269  df-mnf 11270  df-xr 11271  df-ltxr 11272  df-le 11273  df-sub 11467  df-neg 11468  df-div 11896  df-nn 12258  df-2 12327  df-3 12328  df-n0 12529  df-xnn0 12602  df-z 12616  df-uz 12888  df-q 12998  df-rp 13043  df-fz 13562  df-fzo 13710  df-fl 13853  df-mod 13931  df-seq 14066  df-exp 14126  df-fac 14338  df-bc 14367  df-hash 14395  df-cj 15186  df-re 15187  df-im 15188  df-sqrt 15322  df-abs 15323  df-clim 15575  df-sum 15774  df-dvds 16343  df-gcd 16585  df-prm 16762  df-pc 16929  df-sets 17256  df-slot 17274  df-ndx 17286  df-base 17302  df-ress 17323  df-plusg 17355  df-0g 17526  df-mgm 18730  df-sgrp 18821  df-mnd 18837  df-grp 19060  df-minusg 19061  df-subg 19246  df-eqg 19248  df-ga 19417

This theorem is used by:  odcau  19731  slwhash  19751