Theorem sylow1 19810

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 2761	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
8		eqid 2761	. . 3 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}
9		oveq2 7426	. . . . . . 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 7425	. . . . . . 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 2808	. . . . 5 ⊢ (𝑢 = 𝑥 → (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
14	13	rneqd 5920	. . . 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 5920	. . . 4 ⊢ (𝑣 = 𝑦 → ran (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)) = ran (𝑧 ∈ 𝑦 ↦ (𝑥(+g‘𝐺)𝑧)))
17	14, 16	cbvmpov 7513	. . 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 7426	. . . . . . 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 2775	. . . . . 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 3187	. . . . 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 19807	. 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 2761	. . 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 19809	. 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 3170	1 ⊢ (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃↑𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∃wrex 3087  {crab 3413   ⊆ wss 3899  𝒫 cpw 4557  {cpr 4586   class class class wbr 5103  {copab 5167   ↦ cmpt 5186  ran crn 5652  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  [cec 8708  Fincfn 8966   ≤ cle 11337   − cmin 11534  ℕ₀cn0 12599  ↑cexp 14197  ♯chash 14467   ∥ cdvds 16415  ℙcprime 16839   pCnt cpc 17007  Basecbs 17380  +_gcplusg 17421  Grpcgrp 19137  SubGrpcsubg 19323

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-inf2 9635  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

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 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 3366  df-reu 3367  df-rab 3414  df-v 3453  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 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-oadd 8473  df-er 8710  df-ec 8712  df-qs 8716  df-map 8842  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9427  df-inf 9428  df-oi 9497  df-dju 9975  df-card 10013  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-2 12398  df-3 12399  df-n0 12600  df-xnn0 12673  df-z 12687  df-uz 12959  df-q 13069  df-rp 13114  df-fz 13633  df-fzo 13782  df-fl 13925  df-mod 14003  df-seq 14138  df-exp 14198  df-fac 14411  df-bc 14440  df-hash 14468  df-cj 15259  df-re 15260  df-im 15261  df-sqrt 15395  df-abs 15396  df-clim 15648  df-sum 15847  df-dvds 16416  df-gcd 16658  df-prm 16840  df-pc 17008  df-sets 17335  df-slot 17353  df-ndx 17365  df-base 17381  df-ress 17402  df-plusg 17434  df-0g 17605  df-mgm 18809  df-sgrp 18901  df-mnd 18917  df-grp 19140  df-minusg 19141  df-subg 19326  df-eqg 19328  df-ga 19497

This theorem is used by:  odcau  19811  slwhash  19831