Theorem sylow1 19544

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 2737	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
8		eqid 2737	. . 3 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}
9		oveq2 7376	. . . . . . 7 ⊢ (𝑠 = 𝑧 → (𝑢(+g‘𝐺)𝑠) = (𝑢(+g‘𝐺)𝑧))
10	9	cbvmptv 5204	. . . . . 6 ⊢ (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑧))
11		oveq1 7375	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑢(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)𝑧))
12	11	mpteq2dv 5194	. . . . . 6 ⊢ (𝑢 = 𝑥 → (𝑧 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑧)) = (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
13	10, 12	eqtrid 2784	. . . . 5 ⊢ (𝑢 = 𝑥 → (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
14	13	rneqd 5895	. . . 4 ⊢ (𝑢 = 𝑥 → ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = ran (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
15		mpteq1 5189	. . . . 5 ⊢ (𝑣 = 𝑦 → (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)) = (𝑧 ∈ 𝑦 ↦ (𝑥(+g‘𝐺)𝑧)))
16	15	rneqd 5895	. . . 4 ⊢ (𝑣 = 𝑦 → ran (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)) = ran (𝑧 ∈ 𝑦 ↦ (𝑥(+g‘𝐺)𝑧)))
17	14, 16	cbvmpov 7463	. . 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 4694	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → {𝑎, 𝑏} = {𝑥, 𝑦})
19	18	sseq1d 3967	. . . . 5 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↔ {𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}))
20		oveq2 7376	. . . . . . 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 22	. . . . . . 7 ⊢ (𝑏 = 𝑦 → 𝑏 = 𝑦)
22	20, 21	eqeqan12d 2751	. . . . . 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 3162	. . . . 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 633	. . . 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 5173	. . 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 19541	. 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 480	. . 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 480	. . 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 480	. . 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 480	. . 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 480	. . 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 771	. . 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 2737	. . 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 773	. . 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 19543	. 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 3145	1 ⊢ (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃↑𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3401   ⊆ wss 3903  𝒫 cpw 4556  {cpr 4584   class class class wbr 5100  {copab 5162   ↦ cmpt 5181  ran crn 5633  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  [cec 8643  Fincfn 8895   ≤ cle 11179   − cmin 11376  ℕ₀cn0 12413  ↑cexp 13996  ♯chash 14265   ∥ cdvds 16191  ℙcprime 16610   pCnt cpc 16776  Basecbs 17148  +_gcplusg 17189  Grpcgrp 18875  SubGrpcsubg 19062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-disj 5068  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-2o 8408  df-oadd 8411  df-er 8645  df-ec 8647  df-qs 8651  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-sup 9357  df-inf 9358  df-oi 9427  df-dju 9825  df-card 9863  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-div 11807  df-nn 12158  df-2 12220  df-3 12221  df-n0 12414  df-xnn0 12487  df-z 12501  df-uz 12764  df-q 12874  df-rp 12918  df-fz 13436  df-fzo 13583  df-fl 13724  df-mod 13802  df-seq 13937  df-exp 13997  df-fac 14209  df-bc 14238  df-hash 14266  df-cj 15034  df-re 15035  df-im 15036  df-sqrt 15170  df-abs 15171  df-clim 15423  df-sum 15622  df-dvds 16192  df-gcd 16434  df-prm 16611  df-pc 16777  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17149  df-ress 17170  df-plusg 17202  df-0g 17373  df-mgm 18577  df-sgrp 18656  df-mnd 18672  df-grp 18878  df-minusg 18879  df-subg 19065  df-eqg 19067  df-ga 19231

This theorem is referenced by:  odcau  19545  slwhash  19565