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Theorem pgpssslw 19680

Description: Every 𝑃-subgroup is contained in a Sylow 𝑃-subgroup. (Contributed by Mario Carneiro, 16-Jan-2015.)

**Hypotheses**
Ref	Expression
pgpssslw.1	⊢ 𝑋 = (Base‘𝐺)
pgpssslw.2	⊢ 𝑆 = (𝐺 ↾_s 𝐻)
pgpssslw.3	⊢ 𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} ↦ (♯‘𝑥))

**Assertion**
Ref	Expression
pgpssslw	⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻 ⊆ 𝑘)

Distinct variable groups: 𝑥,𝑘,𝑦,𝐺 𝑘,𝐻,𝑥,𝑦 𝑃,𝑘,𝑥,𝑦 𝑘,𝑋,𝑥 𝑘,𝐹 𝑆,𝑘,𝑥,𝑦 Allowed substitution hints: 𝐹(𝑥,𝑦) 𝑋(𝑦)

Proof of Theorem pgpssslw Dummy variables 𝑚 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp2 1153	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑋 ∈ Fin)
2		elrabi 3655	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} → 𝑥 ∈ (SubGrp‘𝐺))
3		pgpssslw.1	. . . . . . . . . . . 12 ⊢ 𝑋 = (Base‘𝐺)
4	3	subgss 19189	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
5	2, 4	syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} → 𝑥 ⊆ 𝑋)
6		ssfi 9153	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Fin ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ Fin)
7	1, 5, 6	syl2an 607	. . . . . . . . 9 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}) → 𝑥 ∈ Fin)
8		hashcl 14388	. . . . . . . . 9 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
9	7, 8	syl 18	. . . . . . . 8 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}) → (♯‘𝑥) ∈ ℕ₀)
10	9	nn0zd 12612	. . . . . . 7 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}) → (♯‘𝑥) ∈ ℤ)
11		pgpssslw.3	. . . . . . 7 ⊢ 𝐹 = (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} ↦ (♯‘𝑥))
12	10, 11	fmptd 7107	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐹:{𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}⟶ℤ)
13	12	frnd 6712	. . . . 5 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℤ)
14		fvex 6892	. . . . . . . 8 ⊢ (♯‘𝑥) ∈ V
15	14, 11	fnmpti 6676	. . . . . . 7 ⊢ 𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}
16		eqimss2 4004	. . . . . . . . . 10 ⊢ (𝑦 = 𝐻 → 𝐻 ⊆ 𝑦)
17	16	biantrud 540	. . . . . . . . 9 ⊢ (𝑦 = 𝐻 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)))
18		oveq2 7416	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐻 → (𝐺 ↾_s 𝑦) = (𝐺 ↾_s 𝐻))
19		pgpssslw.2	. . . . . . . . . . 11 ⊢ 𝑆 = (𝐺 ↾_s 𝐻)
20	18, 19	eqtr4di 2822	. . . . . . . . . 10 ⊢ (𝑦 = 𝐻 → (𝐺 ↾_s 𝑦) = 𝑆)
21	20	breq2d 5122	. . . . . . . . 9 ⊢ (𝑦 = 𝐻 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ 𝑃 pGrp 𝑆))
22	17, 21	bitr3d 284	. . . . . . . 8 ⊢ (𝑦 = 𝐻 → ((𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦) ↔ 𝑃 pGrp 𝑆))
23		simp1 1152	. . . . . . . 8 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ (SubGrp‘𝐺))
24		simp3 1154	. . . . . . . 8 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑃 pGrp 𝑆)
25	22, 23, 24	elrabd 3661	. . . . . . 7 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)})
26		fnfvelrn 7073	. . . . . . 7 ⊢ ((𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} ∧ 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}) → (𝐹‘𝐻) ∈ ran 𝐹)
27	15, 25, 26	sylancr 598	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (𝐹‘𝐻) ∈ ran 𝐹)
28	27	ne0d 4303	. . . . 5 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ≠ ∅)
29		hashcl 14388	. . . . . . . 8 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
30	1, 29	syl 18	. . . . . . 7 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℕ₀)
31	30	nn0red 12562	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℝ)
32		fveq2 6879	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑚 → (♯‘𝑥) = (♯‘𝑚))
33		fvex 6892	. . . . . . . . . . 11 ⊢ (♯‘𝑚) ∈ V
34	32, 11, 33	fvmpt 6987	. . . . . . . . . 10 ⊢ (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} → (𝐹‘𝑚) = (♯‘𝑚))
35	34	adantl 486	. . . . . . . . 9 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}) → (𝐹‘𝑚) = (♯‘𝑚))
36		oveq2 7416	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑚 → (𝐺 ↾_s 𝑦) = (𝐺 ↾_s 𝑚))
37	36	breq2d 5122	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑚 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑚)))
38		sseq2 3971	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑚 → (𝐻 ⊆ 𝑦 ↔ 𝐻 ⊆ 𝑚))
39	37, 38	anbi12d 643	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑚 → ((𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦) ↔ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚)))
40	39	elrab 3659	. . . . . . . . . 10 ⊢ (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} ↔ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚)))
41	1	adantr 485	. . . . . . . . . . . 12 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → 𝑋 ∈ Fin)
42	3	subgss 19189	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (SubGrp‘𝐺) → 𝑚 ⊆ 𝑋)
43	42	ad2antrl 740	. . . . . . . . . . . 12 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → 𝑚 ⊆ 𝑋)
44		ssdomg 8993	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Fin → (𝑚 ⊆ 𝑋 → 𝑚 ≼ 𝑋))
45	41, 43, 44	sylc 66	. . . . . . . . . . 11 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → 𝑚 ≼ 𝑋)
46	41, 43	ssfid 9225	. . . . . . . . . . . 12 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → 𝑚 ∈ Fin)
47		hashdom 14411	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ Fin ∧ 𝑋 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑋) ↔ 𝑚 ≼ 𝑋))
48	46, 41, 47	syl2anc 595	. . . . . . . . . . 11 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → ((♯‘𝑚) ≤ (♯‘𝑋) ↔ 𝑚 ≼ 𝑋))
49	45, 48	mpbird 260	. . . . . . . . . 10 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → (♯‘𝑚) ≤ (♯‘𝑋))
50	40, 49	sylan2b 605	. . . . . . . . 9 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}) → (♯‘𝑚) ≤ (♯‘𝑋))
51	35, 50	eqbrtrd 5134	. . . . . . . 8 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}) → (𝐹‘𝑚) ≤ (♯‘𝑋))
52	51	ralrimiva 3163	. . . . . . 7 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} (𝐹‘𝑚) ≤ (♯‘𝑋))
53		breq1 5113	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑚) → (𝑤 ≤ (♯‘𝑋) ↔ (𝐹‘𝑚) ≤ (♯‘𝑋)))
54	53	ralrn 7081	. . . . . . . 8 ⊢ (𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} → (∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋) ↔ ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} (𝐹‘𝑚) ≤ (♯‘𝑋)))
55	15, 54	ax-mp 5	. . . . . . 7 ⊢ (∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋) ↔ ∀𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} (𝐹‘𝑚) ≤ (♯‘𝑋))
56	52, 55	sylibr 237	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋))
57		brralrspcev 5172	. . . . . 6 ⊢ (((♯‘𝑋) ∈ ℝ ∧ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ 𝑧)
58	31, 56, 57	syl2anc 595	. . . . 5 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ 𝑧)
59		suprzcl 12672	. . . . 5 ⊢ ((ran 𝐹 ⊆ ℤ ∧ ran 𝐹 ≠ ∅ ∧ ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ 𝑧) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
60	13, 28, 58, 59	syl3anc 1396	. . . 4 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
61		fvelrnb 6939	. . . . 5 ⊢ (𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} → (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))
62	15, 61	ax-mp 5	. . . 4 ⊢ (sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ↔ ∃𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < ))
63	60, 62	sylib 221	. . 3 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < ))
64		oveq2 7416	. . . . . 6 ⊢ (𝑦 = 𝑘 → (𝐺 ↾_s 𝑦) = (𝐺 ↾_s 𝑘))
65	64	breq2d 5122	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑘)))
66		sseq2 3971	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝐻 ⊆ 𝑦 ↔ 𝐻 ⊆ 𝑘))
67	65, 66	anbi12d 643	. . . 4 ⊢ (𝑦 = 𝑘 → ((𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦) ↔ (𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘)))
68	67	rexrab 3668	. . 3 ⊢ (∃𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < ) ↔ ∃𝑘 ∈ (SubGrp‘𝐺)((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))
69	63, 68	sylib 221	. 2 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (SubGrp‘𝐺)((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))
70		simpl3 1210	. . . 4 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑃 pGrp 𝑆)
71		pgpprm 19659	. . . 4 ⊢ (𝑃 pGrp 𝑆 → 𝑃 ∈ ℙ)
72	70, 71	syl 18	. . 3 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑃 ∈ ℙ)
73		simprl 782	. . 3 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑘 ∈ (SubGrp‘𝐺))
74		zssre 12594	. . . . . . . . . . . . 13 ⊢ ℤ ⊆ ℝ
75	13, 74	sstrdi 3957	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℝ)
76	75	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → ran 𝐹 ⊆ ℝ)
77	28	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → ran 𝐹 ≠ ∅)
78	58	ad2antrr 738	. . . . . . . . . . 11 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ 𝑧)
79		simprl 782	. . . . . . . . . . . . . 14 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑚 ∈ (SubGrp‘𝐺))
80		simprrr 793	. . . . . . . . . . . . . . 15 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑃 pGrp (𝐺 ↾_s 𝑚))
81		simprrl 792	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) → (𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘))
82	81	adantr 485	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘))
83	82	simprd 500	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝐻 ⊆ 𝑘)
84		simprrl 792	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑘 ⊆ 𝑚)
85	83, 84	sstrd 3955	. . . . . . . . . . . . . . 15 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝐻 ⊆ 𝑚)
86	80, 85	jca 520	. . . . . . . . . . . . . 14 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))
87	39, 79, 86	elrabd 3661	. . . . . . . . . . . . 13 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)})
88	87, 34	syl 18	. . . . . . . . . . . 12 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (𝐹‘𝑚) = (♯‘𝑚))
89		fnfvelrn 7073	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} ∧ 𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}) → (𝐹‘𝑚) ∈ ran 𝐹)
90	15, 87, 89	sylancr 598	. . . . . . . . . . . 12 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (𝐹‘𝑚) ∈ ran 𝐹)
91	88, 90	eqeltrrd 2870	. . . . . . . . . . 11 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (♯‘𝑚) ∈ ran 𝐹)
92	76, 77, 78, 91	suprubd 12173	. . . . . . . . . 10 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (♯‘𝑚) ≤ sup(ran 𝐹, ℝ, < ))
93		simprrr 793	. . . . . . . . . . . 12 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) → (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < ))
94	93	adantr 485	. . . . . . . . . . 11 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < ))
95	73	adantr 485	. . . . . . . . . . . . 13 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑘 ∈ (SubGrp‘𝐺))
96	67, 95, 82	elrabd 3661	. . . . . . . . . . . 12 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)})
97		fveq2 6879	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (♯‘𝑥) = (♯‘𝑘))
98		fvex 6892	. . . . . . . . . . . . 13 ⊢ (♯‘𝑘) ∈ V
99	97, 11, 98	fvmpt 6987	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} → (𝐹‘𝑘) = (♯‘𝑘))
100	96, 99	syl 18	. . . . . . . . . . 11 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (𝐹‘𝑘) = (♯‘𝑘))
101	94, 100	eqtr3d 2806	. . . . . . . . . 10 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → sup(ran 𝐹, ℝ, < ) = (♯‘𝑘))
102	92, 101	breqtrd 5138	. . . . . . . . 9 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (♯‘𝑚) ≤ (♯‘𝑘))
103		simpll2 1230	. . . . . . . . . . 11 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑋 ∈ Fin)
104	42	ad2antrl 740	. . . . . . . . . . 11 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑚 ⊆ 𝑋)
105	103, 104	ssfid 9225	. . . . . . . . . 10 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑚 ∈ Fin)
106	105, 84	ssfid 9225	. . . . . . . . . 10 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑘 ∈ Fin)
107		hashcl 14388	. . . . . . . . . . 11 ⊢ (𝑚 ∈ Fin → (♯‘𝑚) ∈ ℕ₀)
108		hashcl 14388	. . . . . . . . . . 11 ⊢ (𝑘 ∈ Fin → (♯‘𝑘) ∈ ℕ₀)
109		nn0re 12509	. . . . . . . . . . . 12 ⊢ ((♯‘𝑚) ∈ ℕ₀ → (♯‘𝑚) ∈ ℝ)
110		nn0re 12509	. . . . . . . . . . . 12 ⊢ ((♯‘𝑘) ∈ ℕ₀ → (♯‘𝑘) ∈ ℝ)
111		lenlt 11284	. . . . . . . . . . . 12 ⊢ (((♯‘𝑚) ∈ ℝ ∧ (♯‘𝑘) ∈ ℝ) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
112	109, 110, 111	syl2an 607	. . . . . . . . . . 11 ⊢ (((♯‘𝑚) ∈ ℕ₀ ∧ (♯‘𝑘) ∈ ℕ₀) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
113	107, 108, 112	syl2an 607	. . . . . . . . . 10 ⊢ ((𝑚 ∈ Fin ∧ 𝑘 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
114	105, 106, 113	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
115	102, 114	mpbid 235	. . . . . . . 8 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → ¬ (♯‘𝑘) < (♯‘𝑚))
116		php3 9189	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ Fin ∧ 𝑘 ⊊ 𝑚) → 𝑘 ≺ 𝑚)
117	116	ex 417	. . . . . . . . . 10 ⊢ (𝑚 ∈ Fin → (𝑘 ⊊ 𝑚 → 𝑘 ≺ 𝑚))
118	105, 117	syl 18	. . . . . . . . 9 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (𝑘 ⊊ 𝑚 → 𝑘 ≺ 𝑚))
119		hashsdom 14413	. . . . . . . . . 10 ⊢ ((𝑘 ∈ Fin ∧ 𝑚 ∈ Fin) → ((♯‘𝑘) < (♯‘𝑚) ↔ 𝑘 ≺ 𝑚))
120	106, 105, 119	syl2anc 595	. . . . . . . . 9 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → ((♯‘𝑘) < (♯‘𝑚) ↔ 𝑘 ≺ 𝑚))
121	118, 120	sylibrd 262	. . . . . . . 8 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (𝑘 ⊊ 𝑚 → (♯‘𝑘) < (♯‘𝑚)))
122	115, 121	mtod 201	. . . . . . 7 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → ¬ 𝑘 ⊊ 𝑚)
123		sspss 4064	. . . . . . . . 9 ⊢ (𝑘 ⊆ 𝑚 ↔ (𝑘 ⊊ 𝑚 ∨ 𝑘 = 𝑚))
124	84, 123	sylib 221	. . . . . . . 8 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (𝑘 ⊊ 𝑚 ∨ 𝑘 = 𝑚))
125	124	ord 877	. . . . . . 7 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → (¬ 𝑘 ⊊ 𝑚 → 𝑘 = 𝑚))
126	122, 125	mpd 16	. . . . . 6 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)))) → 𝑘 = 𝑚)
127	126	expr 461	. . . . 5 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ 𝑚 ∈ (SubGrp‘𝐺)) → ((𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)) → 𝑘 = 𝑚))
128	81	simpld 499	. . . . . . 7 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑃 pGrp (𝐺 ↾_s 𝑘))
129	128	adantr 485	. . . . . 6 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ 𝑚 ∈ (SubGrp‘𝐺)) → 𝑃 pGrp (𝐺 ↾_s 𝑘))
130		oveq2 7416	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝐺 ↾_s 𝑘) = (𝐺 ↾_s 𝑚))
131	130	breq2d 5122	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (𝑃 pGrp (𝐺 ↾_s 𝑘) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑚)))
132		eqimss 4003	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → 𝑘 ⊆ 𝑚)
133	132	biantrurd 541	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (𝑃 pGrp (𝐺 ↾_s 𝑚) ↔ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚))))
134	131, 133	bitrd 282	. . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑃 pGrp (𝐺 ↾_s 𝑘) ↔ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚))))
135	129, 134	syl5ibcom 248	. . . . 5 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ 𝑚 ∈ (SubGrp‘𝐺)) → (𝑘 = 𝑚 → (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚))))
136	127, 135	impbid 215	. . . 4 ⊢ ((((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) ∧ 𝑚 ∈ (SubGrp‘𝐺)) → ((𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)) ↔ 𝑘 = 𝑚))
137	136	ralrimiva 3163	. . 3 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) → ∀𝑚 ∈ (SubGrp‘𝐺)((𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)) ↔ 𝑘 = 𝑚))
138		isslw 19674	. . 3 ⊢ (𝑘 ∈ (𝑃 pSyl 𝐺) ↔ (𝑃 ∈ ℙ ∧ 𝑘 ∈ (SubGrp‘𝐺) ∧ ∀𝑚 ∈ (SubGrp‘𝐺)((𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚)) ↔ 𝑘 = 𝑚)))
139	72, 73, 137, 138	syl3anbrc 1360	. 2 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝑘 ∈ (𝑃 pSyl 𝐺))
140	81	simprd 500	. 2 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑘 ∈ (SubGrp‘𝐺) ∧ ((𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘) ∧ (𝐹‘𝑘) = sup(ran 𝐹, ℝ, < )))) → 𝐻 ⊆ 𝑘)
141	69, 139, 140	reximssdv 3189	1 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑘 ∈ (𝑃 pSyl 𝐺)𝐻 ⊆ 𝑘)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 ∀wral 3085 ∃wrex 3095 {crab 3423 ⊆ wss 3913 ⊊ wpss 3914 ∅c0 4294 class class class wbr 5110 ↦ cmpt 5193 ran crn 5660 Fn wfn 6528 ‘cfv 6533 (class class class)co 7408 ≼ cdom 8937 ≺ csdm 8938 Fincfn 8939 supcsup 9396 ℝcr 11095 < clt 11239 ≤ cle 11240 ℕ₀cn0 12500 ℤcz 12587 ♯chash 14362 ℙcprime 16725 Basecbs 17265 ↾_s cress 17286 SubGrpcsubg 19182 pGrp cpgp 19592 pSyl cslw 19593

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-pre-sup 11174

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 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-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-om 7859 df-1st 7982 df-2nd 7983 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-card 9921 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-n0 12501 df-xnn0 12574 df-z 12588 df-uz 12859 df-fz 13532 df-hash 14363 df-subg 19185 df-pgp 19596 df-slw 19597

This theorem is referenced by: slwn0 19681

W3C validator