Theorem sylow1 17841

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 2610	. . 3 ⊢ (+g‘𝐺) = (+g‘𝐺)
8		eqid 2610	. . 3 ⊢ {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)} = {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)}
9		oveq2 6557	. . . . . . 7 ⊢ (𝑠 = 𝑧 → (𝑢(+g‘𝐺)𝑠) = (𝑢(+g‘𝐺)𝑧))
10	9	cbvmptv 4678	. . . . . 6 ⊢ (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑧))
11		oveq1 6556	. . . . . . 7 ⊢ (𝑢 = 𝑥 → (𝑢(+g‘𝐺)𝑧) = (𝑥(+g‘𝐺)𝑧))
12	11	mpteq2dv 4673	. . . . . 6 ⊢ (𝑢 = 𝑥 → (𝑧 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑧)) = (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
13	10, 12	syl5eq 2656	. . . . 5 ⊢ (𝑢 = 𝑥 → (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
14	13	rneqd 5274	. . . 4 ⊢ (𝑢 = 𝑥 → ran (𝑠 ∈ 𝑣 ↦ (𝑢(+g‘𝐺)𝑠)) = ran (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)))
15		mpteq1 4665	. . . . 5 ⊢ (𝑣 = 𝑦 → (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)) = (𝑧 ∈ 𝑦 ↦ (𝑥(+g‘𝐺)𝑧)))
16	15	rneqd 5274	. . . 4 ⊢ (𝑣 = 𝑦 → ran (𝑧 ∈ 𝑣 ↦ (𝑥(+g‘𝐺)𝑧)) = ran (𝑧 ∈ 𝑦 ↦ (𝑥(+g‘𝐺)𝑧)))
17	14, 16	cbvmpt2v 6633	. . 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 4214	. . . . . 6 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → {𝑎, 𝑏} = {𝑥, 𝑦})
19	18	sseq1d 3595	. . . . 5 ⊢ ((𝑎 = 𝑥 ∧ 𝑏 = 𝑦) → ({𝑎, 𝑏} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)} ↔ {𝑥, 𝑦} ⊆ {𝑠 ∈ 𝒫 𝑋 ∣ (#‘𝑠) = (𝑃↑𝑁)}))
20		oveq2 6557	. . . . . . 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 2626	. . . . . 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 3034	. . . . 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 743	. . . 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 4654	. . 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 17838	. 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 790	. . 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 2610	. . 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 792	. . 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 17840	. 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 3017	1 ⊢ (𝜑 → ∃𝑔 ∈ (SubGrp‘𝐺)(#‘𝑔) = (𝑃↑𝑁))

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {crab 2900   ⊆ wss 3540  𝒫 cpw 4108  {cpr 4127   class class class wbr 4583  {copab 4642   ↦ cmpt 4643  ran crn 5039  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  [cec 7627  Fincfn 7841   ≤ cle 9954   − cmin 10145  ℕ₀cn0 11169  ↑cexp 12722  #chash 12979   ∥ cdvds 14821  ℙcprime 15223   pCnt cpc 15379  Basecbs 15695  +_gcplusg 15768  Grpcgrp 17245  SubGrpcsubg 17411

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-ec 7631  df-qs 7635  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-fac 12923  df-bc 12952  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-gcd 15055  df-prm 15224  df-pc 15380  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-0g 15925  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-grp 17248  df-minusg 17249  df-subg 17414  df-eqg 17416  df-ga 17546

This theorem is referenced by:  odcau  17842  slwhash  17862