Theorem sylow1 19578

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 2736	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
8		eqid 2736	. . 3 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} = {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}
9		oveq2 7375	. . . . . . 7 ⊢ (𝑠 = 𝑧 → (𝑢(+g‘𝐺)𝑠) = (𝑢(+g‘𝐺)𝑧))
10	9	cbvmptv 5189	. . . . . 6 ⊢ (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑧))
11		oveq1 7374	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑢(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)𝑧))
12	11	mpteq2dv 5179	. . . . . 6 ⊢ (𝑢 = 𝑥 → (𝑧 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑧)) = (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
13	10, 12	eqtrid 2783	. . . . 5 ⊢ (𝑢 = 𝑥 → (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
14	13	rneqd 5893	. . . 4 ⊢ (𝑢 = 𝑥 → ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = ran (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
15		mpteq1 5174	. . . . 5 ⊢ (𝑣 = 𝑦 → (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)) = (𝑧 ∈ 𝑦 ↦ (𝑥(+g‘𝐺)𝑧)))
16	15	rneqd 5893	. . . 4 ⊢ (𝑣 = 𝑦 → ran (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)) = ran (𝑧 ∈ 𝑦 ↦ (𝑥(+g‘𝐺)𝑧)))
17	14, 16	cbvmpov 7462	. . 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 4679	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → {𝑎, 𝑏} = {𝑥, 𝑦})
19	18	sseq1d 3953	. . . . 5 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)} ↔ {𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (♯‘𝑠) = (𝑃↑𝑁)}))
20		oveq2 7375	. . . . . . 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 2750	. . . . . 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 3161	. . . . 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 5158	. . 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 19575	. 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 2736	. . 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 19577	. 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 3144	1 ⊢ (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(♯‘𝑔) = (𝑃↑𝑁))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  {crab 3389   ⊆ wss 3889  𝒫 cpw 4541  {cpr 4569   class class class wbr 5085  {copab 5147   ↦ cmpt 5166  ran crn 5632  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  [cec 8641  Fincfn 8893   ≤ cle 11180   − cmin 11377  ℕ₀cn0 12437  ↑cexp 14023  ♯chash 14292   ∥ cdvds 16221  ℙcprime 16640   pCnt cpc 16807  Basecbs 17179  +_gcplusg 17220  Grpcgrp 18909  SubGrpcsubg 19096

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-disj 5053  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-oadd 8409  df-er 8643  df-ec 8645  df-qs 8649  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-oi 9425  df-dju 9825  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-n0 12438  df-xnn0 12511  df-z 12525  df-uz 12789  df-q 12899  df-rp 12943  df-fz 13462  df-fzo 13609  df-fl 13751  df-mod 13829  df-seq 13964  df-exp 14024  df-fac 14236  df-bc 14265  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-clim 15450  df-sum 15649  df-dvds 16222  df-gcd 16464  df-prm 16641  df-pc 16808  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912  df-minusg 18913  df-subg 19099  df-eqg 19101  df-ga 19265

This theorem is referenced by:  odcau  19579  slwhash  19599