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Theorem pgpssslw 19647

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 1136	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑋 ∈ Fin)
2		elrabi 3690	. . . . . . . . . . 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 19158	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (SubGrp‘𝐺) → 𝑥 ⊆ 𝑋)
5	2, 4	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} → 𝑥 ⊆ 𝑋)
6		ssfi 9212	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Fin ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ Fin)
7	1, 5, 6	syl2an 596	. . . . . . . . 9 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}) → 𝑥 ∈ Fin)
8		hashcl 14392	. . . . . . . . 9 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
9	7, 8	syl 17	. . . . . . . 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 12637	. . . . . . 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 7134	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐹:{𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}⟶ℤ)
13	12	frnd 6745	. . . . 5 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℤ)
14		fvex 6920	. . . . . . . 8 ⊢ (♯‘𝑥) ∈ V
15	14, 11	fnmpti 6712	. . . . . . 7 ⊢ 𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}
16		eqimss2 4055	. . . . . . . . . 10 ⊢ (𝑦 = 𝐻 → 𝐻 ⊆ 𝑦)
17	16	biantrud 531	. . . . . . . . 9 ⊢ (𝑦 = 𝐻 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)))
18		oveq2 7439	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐻 → (𝐺 ↾_s 𝑦) = (𝐺 ↾_s 𝐻))
19		pgpssslw.2	. . . . . . . . . . 11 ⊢ 𝑆 = (𝐺 ↾_s 𝐻)
20	18, 19	eqtr4di 2793	. . . . . . . . . 10 ⊢ (𝑦 = 𝐻 → (𝐺 ↾_s 𝑦) = 𝑆)
21	20	breq2d 5160	. . . . . . . . 9 ⊢ (𝑦 = 𝐻 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ 𝑃 pGrp 𝑆))
22	17, 21	bitr3d 281	. . . . . . . 8 ⊢ (𝑦 = 𝐻 → ((𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦) ↔ 𝑃 pGrp 𝑆))
23		simp1 1135	. . . . . . . 8 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ (SubGrp‘𝐺))
24		simp3 1137	. . . . . . . 8 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑃 pGrp 𝑆)
25	22, 23, 24	elrabd 3697	. . . . . . 7 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)})
26		fnfvelrn 7100	. . . . . . 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 587	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (𝐹‘𝐻) ∈ ran 𝐹)
28	27	ne0d 4348	. . . . 5 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ≠ ∅)
29		hashcl 14392	. . . . . . . 8 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
30	1, 29	syl 17	. . . . . . 7 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℕ₀)
31	30	nn0red 12586	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℝ)
32		fveq2 6907	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑚 → (♯‘𝑥) = (♯‘𝑚))
33		fvex 6920	. . . . . . . . . . 11 ⊢ (♯‘𝑚) ∈ V
34	32, 11, 33	fvmpt 7016	. . . . . . . . . 10 ⊢ (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} → (𝐹‘𝑚) = (♯‘𝑚))
35	34	adantl 481	. . . . . . . . 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 7439	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑚 → (𝐺 ↾_s 𝑦) = (𝐺 ↾_s 𝑚))
37	36	breq2d 5160	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑚 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑚)))
38		sseq2 4022	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑚 → (𝐻 ⊆ 𝑦 ↔ 𝐻 ⊆ 𝑚))
39	37, 38	anbi12d 632	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑚 → ((𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦) ↔ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚)))
40	39	elrab 3695	. . . . . . . . . 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 480	. . . . . . . . . . . 12 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → 𝑋 ∈ Fin)
42	3	subgss 19158	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (SubGrp‘𝐺) → 𝑚 ⊆ 𝑋)
43	42	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → 𝑚 ⊆ 𝑋)
44		ssdomg 9039	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ Fin → (𝑚 ⊆ 𝑋 → 𝑚 ≼ 𝑋))
45	41, 43, 44	sylc 65	. . . . . . . . . . 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 9299	. . . . . . . . . . . 12 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → 𝑚 ∈ Fin)
47		hashdom 14415	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ Fin ∧ 𝑋 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑋) ↔ 𝑚 ≼ 𝑋))
48	46, 41, 47	syl2anc 584	. . . . . . . . . . 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 257	. . . . . . . . . 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 594	. . . . . . . . 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 5170	. . . . . . . 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 3144	. . . . . . 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 5151	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑚) → (𝑤 ≤ (♯‘𝑋) ↔ (𝐹‘𝑚) ≤ (♯‘𝑋)))
54	53	ralrn 7108	. . . . . . . 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 234	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋))
57		brralrspcev 5208	. . . . . 6 ⊢ (((♯‘𝑋) ∈ ℝ ∧ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ 𝑧)
58	31, 56, 57	syl2anc 584	. . . . 5 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ 𝑧)
59		suprzcl 12696	. . . . 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 1370	. . . 4 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
61		fvelrnb 6969	. . . . 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 218	. . 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 7439	. . . . . 6 ⊢ (𝑦 = 𝑘 → (𝐺 ↾_s 𝑦) = (𝐺 ↾_s 𝑘))
65	64	breq2d 5160	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑘)))
66		sseq2 4022	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝐻 ⊆ 𝑦 ↔ 𝐻 ⊆ 𝑘))
67	65, 66	anbi12d 632	. . . 4 ⊢ (𝑦 = 𝑘 → ((𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦) ↔ (𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘)))
68	67	rexrab 3705	. . 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 218	. 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 1192	. . . 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 19626	. . . 4 ⊢ (𝑃 pGrp 𝑆 → 𝑃 ∈ ℙ)
72	70, 71	syl 17	. . 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 771	. . 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 12618	. . . . . . . . . . . . 13 ⊢ ℤ ⊆ ℝ
75	13, 74	sstrdi 4008	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℝ)
76	75	ad2antrr 726	. . . . . . . . . . 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 726	. . . . . . . . . . 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 726	. . . . . . . . . . 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 771	. . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . 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 4006	. . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . 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 3697	. . . . . . . . . . . . 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 17	. . . . . . . . . . . 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 7100	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 2840	. . . . . . . . . . 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 12228	. . . . . . . . . 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 782	. . . . . . . . . . . 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 480	. . . . . . . . . . 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 480	. . . . . . . . . . . . 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 3697	. . . . . . . . . . . 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 6907	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (♯‘𝑥) = (♯‘𝑘))
98		fvex 6920	. . . . . . . . . . . . 13 ⊢ (♯‘𝑘) ∈ V
99	97, 11, 98	fvmpt 7016	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} → (𝐹‘𝑘) = (♯‘𝑘))
100	96, 99	syl 17	. . . . . . . . . . 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 2777	. . . . . . . . . 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 5174	. . . . . . . . 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 1212	. . . . . . . . . . 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 728	. . . . . . . . . . 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 9299	. . . . . . . . . 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 9299	. . . . . . . . . 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 14392	. . . . . . . . . . 11 ⊢ (𝑚 ∈ Fin → (♯‘𝑚) ∈ ℕ₀)
108		hashcl 14392	. . . . . . . . . . 11 ⊢ (𝑘 ∈ Fin → (♯‘𝑘) ∈ ℕ₀)
109		nn0re 12533	. . . . . . . . . . . 12 ⊢ ((♯‘𝑚) ∈ ℕ₀ → (♯‘𝑚) ∈ ℝ)
110		nn0re 12533	. . . . . . . . . . . 12 ⊢ ((♯‘𝑘) ∈ ℕ₀ → (♯‘𝑘) ∈ ℝ)
111		lenlt 11337	. . . . . . . . . . . 12 ⊢ (((♯‘𝑚) ∈ ℝ ∧ (♯‘𝑘) ∈ ℝ) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
112	109, 110, 111	syl2an 596	. . . . . . . . . . 11 ⊢ (((♯‘𝑚) ∈ ℕ₀ ∧ (♯‘𝑘) ∈ ℕ₀) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
113	107, 108, 112	syl2an 596	. . . . . . . . . 10 ⊢ ((𝑚 ∈ Fin ∧ 𝑘 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
114	105, 106, 113	syl2anc 584	. . . . . . . . 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 232	. . . . . . . 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 9247	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ Fin ∧ 𝑘 ⊊ 𝑚) → 𝑘 ≺ 𝑚)
117	116	ex 412	. . . . . . . . . 10 ⊢ (𝑚 ∈ Fin → (𝑘 ⊊ 𝑚 → 𝑘 ≺ 𝑚))
118	105, 117	syl 17	. . . . . . . . 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 14417	. . . . . . . . . 10 ⊢ ((𝑘 ∈ Fin ∧ 𝑚 ∈ Fin) → ((♯‘𝑘) < (♯‘𝑚) ↔ 𝑘 ≺ 𝑚))
120	106, 105, 119	syl2anc 584	. . . . . . . . 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 259	. . . . . . . 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 198	. . . . . . 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 4112	. . . . . . . . 9 ⊢ (𝑘 ⊆ 𝑚 ↔ (𝑘 ⊊ 𝑚 ∨ 𝑘 = 𝑚))
124	84, 123	sylib 218	. . . . . . . 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 864	. . . . . . 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 15	. . . . . 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 456	. . . . 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 494	. . . . . . 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 480	. . . . . 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 7439	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝐺 ↾_s 𝑘) = (𝐺 ↾_s 𝑚))
131	130	breq2d 5160	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (𝑃 pGrp (𝐺 ↾_s 𝑘) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑚)))
132		eqimss 4054	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → 𝑘 ⊆ 𝑚)
133	132	biantrurd 532	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (𝑃 pGrp (𝐺 ↾_s 𝑚) ↔ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚))))
134	131, 133	bitrd 279	. . . . . 6 ⊢ (𝑘 = 𝑚 → (𝑃 pGrp (𝐺 ↾_s 𝑘) ↔ (𝑘 ⊆ 𝑚 ∧ 𝑃 pGrp (𝐺 ↾_s 𝑚))))
135	129, 134	syl5ibcom 245	. . . . 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 212	. . . 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 3144	. . 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 19641	. . 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 1342	. 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 495	. 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 3171	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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∃wrex 3068 {crab 3433 ⊆ wss 3963 ⊊ wpss 3964 ∅c0 4339 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 ≼ cdom 8982 ≺ csdm 8983 Fincfn 8984 supcsup 9478 ℝcr 11152 < clt 11293 ≤ cle 11294 ℕ₀cn0 12524 ℤcz 12611 ♯chash 14366 ℙcprime 16705 Basecbs 17245 ↾_s cress 17274 SubGrpcsubg 19151 pGrp cpgp 19559 pSyl cslw 19560

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-xnn0 12598 df-z 12612 df-uz 12877 df-fz 13545 df-hash 14367 df-subg 19154 df-pgp 19563 df-slw 19564

This theorem is referenced by: slwn0 19648

W3C validator