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Theorem pgpssslw 18957

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 1139	. . . . . . . . . 10 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑋 ∈ Fin)
2		elrabi 3585	. . . . . . . . . . 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 18498	. . . . . . . . . . 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 8829	. . . . . . . . . 10 ⊢ ((𝑋 ∈ Fin ∧ 𝑥 ⊆ 𝑋) → 𝑥 ∈ Fin)
7	1, 5, 6	syl2an 599	. . . . . . . . 9 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ 𝑥 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}) → 𝑥 ∈ Fin)
8		hashcl 13888	. . . . . . . . 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 12245	. . . . . . 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 6909	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐹:{𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}⟶ℤ)
13	12	frnd 6531	. . . . 5 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ⊆ ℤ)
14		fvex 6708	. . . . . . . 8 ⊢ (♯‘𝑥) ∈ V
15	14, 11	fnmpti 6499	. . . . . . 7 ⊢ 𝐹 Fn {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)}
16		eqimss2 3944	. . . . . . . . . 10 ⊢ (𝑦 = 𝐻 → 𝐻 ⊆ 𝑦)
17	16	biantrud 535	. . . . . . . . 9 ⊢ (𝑦 = 𝐻 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)))
18		oveq2 7199	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐻 → (𝐺 ↾_s 𝑦) = (𝐺 ↾_s 𝐻))
19		pgpssslw.2	. . . . . . . . . . 11 ⊢ 𝑆 = (𝐺 ↾_s 𝐻)
20	18, 19	eqtr4di 2789	. . . . . . . . . 10 ⊢ (𝑦 = 𝐻 → (𝐺 ↾_s 𝑦) = 𝑆)
21	20	breq2d 5051	. . . . . . . . 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 1138	. . . . . . . 8 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ (SubGrp‘𝐺))
24		simp3 1140	. . . . . . . 8 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝑃 pGrp 𝑆)
25	22, 23, 24	elrabd 3593	. . . . . . 7 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → 𝐻 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)})
26		fnfvelrn 6879	. . . . . . 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 590	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (𝐹‘𝐻) ∈ ran 𝐹)
28	27	ne0d 4236	. . . . 5 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ran 𝐹 ≠ ∅)
29		hashcl 13888	. . . . . . . 8 ⊢ (𝑋 ∈ Fin → (♯‘𝑋) ∈ ℕ₀)
30	1, 29	syl 17	. . . . . . 7 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℕ₀)
31	30	nn0red 12116	. . . . . 6 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → (♯‘𝑋) ∈ ℝ)
32		fveq2 6695	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑚 → (♯‘𝑥) = (♯‘𝑚))
33		fvex 6708	. . . . . . . . . . 11 ⊢ (♯‘𝑚) ∈ V
34	32, 11, 33	fvmpt 6796	. . . . . . . . . 10 ⊢ (𝑚 ∈ {𝑦 ∈ (SubGrp‘𝐺) ∣ (𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦)} → (𝐹‘𝑚) = (♯‘𝑚))
35	34	adantl 485	. . . . . . . . 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 7199	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑚 → (𝐺 ↾_s 𝑦) = (𝐺 ↾_s 𝑚))
37	36	breq2d 5051	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑚 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑚)))
38		sseq2 3913	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑚 → (𝐻 ⊆ 𝑦 ↔ 𝐻 ⊆ 𝑚))
39	37, 38	anbi12d 634	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑚 → ((𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦) ↔ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚)))
40	39	elrab 3591	. . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → 𝑋 ∈ Fin)
42	3	subgss 18498	. . . . . . . . . . . . 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 8652	. . . . . . . . . . . 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 8876	. . . . . . . . . . . 12 ⊢ (((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) ∧ (𝑚 ∈ (SubGrp‘𝐺) ∧ (𝑃 pGrp (𝐺 ↾_s 𝑚) ∧ 𝐻 ⊆ 𝑚))) → 𝑚 ∈ Fin)
47		hashdom 13911	. . . . . . . . . . . 12 ⊢ ((𝑚 ∈ Fin ∧ 𝑋 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑋) ↔ 𝑚 ≼ 𝑋))
48	46, 41, 47	syl2anc 587	. . . . . . . . . . 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 597	. . . . . . . . 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 5061	. . . . . . . 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 3095	. . . . . . 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 5042	. . . . . . . . 9 ⊢ (𝑤 = (𝐹‘𝑚) → (𝑤 ≤ (♯‘𝑋) ↔ (𝐹‘𝑚) ≤ (♯‘𝑋)))
54	53	ralrn 6885	. . . . . . . 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 5099	. . . . . 6 ⊢ (((♯‘𝑋) ∈ ℝ ∧ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ (♯‘𝑋)) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ 𝑧)
58	31, 56, 57	syl2anc 587	. . . . 5 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → ∃𝑧 ∈ ℝ ∀𝑤 ∈ ran 𝐹 𝑤 ≤ 𝑧)
59		suprzcl 12222	. . . . 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 1373	. . . 4 ⊢ ((𝐻 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ Fin ∧ 𝑃 pGrp 𝑆) → sup(ran 𝐹, ℝ, < ) ∈ ran 𝐹)
61		fvelrnb 6751	. . . . 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 7199	. . . . . 6 ⊢ (𝑦 = 𝑘 → (𝐺 ↾_s 𝑦) = (𝐺 ↾_s 𝑘))
65	64	breq2d 5051	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑃 pGrp (𝐺 ↾_s 𝑦) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑘)))
66		sseq2 3913	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝐻 ⊆ 𝑦 ↔ 𝐻 ⊆ 𝑘))
67	65, 66	anbi12d 634	. . . 4 ⊢ (𝑦 = 𝑘 → ((𝑃 pGrp (𝐺 ↾_s 𝑦) ∧ 𝐻 ⊆ 𝑦) ↔ (𝑃 pGrp (𝐺 ↾_s 𝑘) ∧ 𝐻 ⊆ 𝑘)))
68	67	rexrab 3598	. . 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 1195	. . . 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 18936	. . . 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 12148	. . . . . . . . . . . . 13 ⊢ ℤ ⊆ ℝ
75	13, 74	sstrdi 3899	. . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . . 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 3897	. . . . . . . . . . . . . . 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 515	. . . . . . . . . . . . . 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 3593	. . . . . . . . . . . . 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 6879	. . . . . . . . . . . . 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 590	. . . . . . . . . . . 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 2832	. . . . . . . . . . 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 11759	. . . . . . . . . 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 484	. . . . . . . . . . 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 484	. . . . . . . . . . . . 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 3593	. . . . . . . . . . . 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 6695	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑘 → (♯‘𝑥) = (♯‘𝑘))
98		fvex 6708	. . . . . . . . . . . . 13 ⊢ (♯‘𝑘) ∈ V
99	97, 11, 98	fvmpt 6796	. . . . . . . . . . . 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 2773	. . . . . . . . . 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 5065	. . . . . . . . 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 1215	. . . . . . . . . . 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 8876	. . . . . . . . . 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 8876	. . . . . . . . . 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 13888	. . . . . . . . . . 11 ⊢ (𝑚 ∈ Fin → (♯‘𝑚) ∈ ℕ₀)
108		hashcl 13888	. . . . . . . . . . 11 ⊢ (𝑘 ∈ Fin → (♯‘𝑘) ∈ ℕ₀)
109		nn0re 12064	. . . . . . . . . . . 12 ⊢ ((♯‘𝑚) ∈ ℕ₀ → (♯‘𝑚) ∈ ℝ)
110		nn0re 12064	. . . . . . . . . . . 12 ⊢ ((♯‘𝑘) ∈ ℕ₀ → (♯‘𝑘) ∈ ℝ)
111		lenlt 10876	. . . . . . . . . . . 12 ⊢ (((♯‘𝑚) ∈ ℝ ∧ (♯‘𝑘) ∈ ℝ) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
112	109, 110, 111	syl2an 599	. . . . . . . . . . 11 ⊢ (((♯‘𝑚) ∈ ℕ₀ ∧ (♯‘𝑘) ∈ ℕ₀) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
113	107, 108, 112	syl2an 599	. . . . . . . . . 10 ⊢ ((𝑚 ∈ Fin ∧ 𝑘 ∈ Fin) → ((♯‘𝑚) ≤ (♯‘𝑘) ↔ ¬ (♯‘𝑘) < (♯‘𝑚)))
114	105, 106, 113	syl2anc 587	. . . . . . . . 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 8810	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ Fin ∧ 𝑘 ⊊ 𝑚) → 𝑘 ≺ 𝑚)
117	116	ex 416	. . . . . . . . . 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 13913	. . . . . . . . . 10 ⊢ ((𝑘 ∈ Fin ∧ 𝑚 ∈ Fin) → ((♯‘𝑘) < (♯‘𝑚) ↔ 𝑘 ≺ 𝑚))
120	106, 105, 119	syl2anc 587	. . . . . . . . 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 4000	. . . . . . . . 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 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 460	. . . . 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 498	. . . . . . 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 484	. . . . . 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 7199	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → (𝐺 ↾_s 𝑘) = (𝐺 ↾_s 𝑚))
131	130	breq2d 5051	. . . . . . 7 ⊢ (𝑘 = 𝑚 → (𝑃 pGrp (𝐺 ↾_s 𝑘) ↔ 𝑃 pGrp (𝐺 ↾_s 𝑚)))
132		eqimss 3943	. . . . . . . 8 ⊢ (𝑘 = 𝑚 → 𝑘 ⊆ 𝑚)
133	132	biantrurd 536	. . . . . . 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 3095	. . 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 18951	. . 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 1345	. 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 499	. 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 3185	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 399 ∨ wo 847 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∃wrex 3052 {crab 3055 ⊆ wss 3853 ⊊ wpss 3854 ∅c0 4223 class class class wbr 5039 ↦ cmpt 5120 ran crn 5537 Fn wfn 6353 ‘cfv 6358 (class class class)co 7191 ≼ cdom 8602 ≺ csdm 8603 Fincfn 8604 supcsup 9034 ℝcr 10693 < clt 10832 ≤ cle 10833 ℕ₀cn0 12055 ℤcz 12141 ♯chash 13861 ℙcprime 16191 Basecbs 16666 ↾_s cress 16667 SubGrpcsubg 18491 pGrp cpgp 18872 pSyl cslw 18873

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-oadd 8184 df-er 8369 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-card 9520 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-nn 11796 df-n0 12056 df-xnn0 12128 df-z 12142 df-uz 12404 df-fz 13061 df-hash 13862 df-subg 18494 df-pgp 18876 df-slw 18877

This theorem is referenced by: slwn0 18958

W3C validator