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Theorem gausslemma2dlem1f1o 15933

Description: Lemma for gausslemma2dlem1 15934. (Contributed by Jim Kingdon, 9-Aug-2025.)

**Hypotheses**
Ref	Expression
gausslemma2d.p	⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
gausslemma2d.h	⊢ 𝐻 = ((𝑃 − 1) / 2)
gausslemma2d.r	⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))

**Assertion**
Ref	Expression
gausslemma2dlem1f1o	⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻))

Distinct variable groups: 𝑥,𝐻 𝑥,𝑃 𝜑,𝑥 Allowed substitution hint: 𝑅(𝑥)

Proof of Theorem gausslemma2dlem1f1o Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		gausslemma2d.p	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃 ∈ (ℙ ∖ {2}))
2	1	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝐻)) → 𝑃 ∈ (ℙ ∖ {2}))
3		gausslemma2d.h	. . . . . . . . 9 ⊢ 𝐻 = ((𝑃 − 1) / 2)
4		gausslemma2d.r	. . . . . . . . 9 ⊢ 𝑅 = (𝑥 ∈ (1...𝐻) ↦ if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))))
5		simpr 110	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝐻)) → 𝑥 ∈ (1...𝐻))
6	2, 3, 4, 5	gausslemma2dlem1cl 15932	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝐻)) → if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))) ∈ ℤ)
7	6	ralrimiva 2615	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (1...𝐻)if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))) ∈ ℤ)
8	4	fnmpt 5485	. . . . . . 7 ⊢ (∀𝑥 ∈ (1...𝐻)if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))) ∈ ℤ → 𝑅 Fn (1...𝐻))
9	7, 8	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn (1...𝐻))
10		dffn4 5596	. . . . . 6 ⊢ (𝑅 Fn (1...𝐻) ↔ 𝑅:(1...𝐻)–onto→ran 𝑅)
11	9, 10	sylib 122	. . . . 5 ⊢ (𝜑 → 𝑅:(1...𝐻)–onto→ran 𝑅)
12	1, 3, 4	gausslemma2dlem1a 15931	. . . . . 6 ⊢ (𝜑 → ran 𝑅 = (1...𝐻))
13		foeq3 5588	. . . . . 6 ⊢ (ran 𝑅 = (1...𝐻) → (𝑅:(1...𝐻)–onto→ran 𝑅 ↔ 𝑅:(1...𝐻)–onto→(1...𝐻)))
14	12, 13	syl 14	. . . . 5 ⊢ (𝜑 → (𝑅:(1...𝐻)–onto→ran 𝑅 ↔ 𝑅:(1...𝐻)–onto→(1...𝐻)))
15	11, 14	mpbid 147	. . . 4 ⊢ (𝜑 → 𝑅:(1...𝐻)–onto→(1...𝐻))
16		fof 5590	. . . 4 ⊢ (𝑅:(1...𝐻)–onto→(1...𝐻) → 𝑅:(1...𝐻)⟶(1...𝐻))
17	15, 16	syl 14	. . 3 ⊢ (𝜑 → 𝑅:(1...𝐻)⟶(1...𝐻))
18		simprl 531	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 𝑦 ∈ (1...𝐻))
19	18	elfzelzd 10360	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 𝑦 ∈ ℤ)
20	19	adantr 276	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → 𝑦 ∈ ℤ)
21	20	adantr 276	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 𝑦 ∈ ℤ)
22	21	zcnd 9701	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 𝑦 ∈ ℂ)
23		simprr 533	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 𝑧 ∈ (1...𝐻))
24	23	elfzelzd 10360	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 𝑧 ∈ ℤ)
25	24	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 𝑧 ∈ ℤ)
26	25	zcnd 9701	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 𝑧 ∈ ℂ)
27		2cnd 9310	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 2 ∈ ℂ)
28		2ap0 9330	. . . . . . . 8 ⊢ 2 # 0
29	28	a1i 9	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 2 # 0)
30		simplr 529	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑦) = (𝑅‘𝑧))
31		oveq1 6057	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 · 2) = (𝑦 · 2))
32	31	breq1d 4119	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑦 · 2) < (𝑃 / 2)))
33	31	oveq2d 6066	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑦 · 2)))
34	32, 31, 33	ifbieq12d 3649	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))) = if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2))))
35	1	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 𝑃 ∈ (ℙ ∖ {2}))
36	35, 3, 4, 18	gausslemma2dlem1cl 15932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2))) ∈ ℤ)
37	4, 34, 18, 36	fvmptd3 5771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → (𝑅‘𝑦) = if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2))))
38	37	ad2antrr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑦) = if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2))))
39		simpr 110	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑦 · 2) < (𝑃 / 2))
40	39	iftrued 3629	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2))) = (𝑦 · 2))
41	38, 40	eqtrd 2265	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑦) = (𝑦 · 2))
42		oveq1 6057	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 · 2) = (𝑧 · 2))
43	42	breq1d 4119	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑧 · 2) < (𝑃 / 2)))
44	42	oveq2d 6066	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑧 · 2)))
45	43, 42, 44	ifbieq12d 3649	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))) = if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))))
46	35, 3, 4, 23	gausslemma2dlem1cl 15932	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))) ∈ ℤ)
47	4, 45, 23, 46	fvmptd3 5771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → (𝑅‘𝑧) = if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))))
48	47	ad2antrr 488	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑧) = if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))))
49		2z 9605	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
50		dvdsmul2 12500	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (𝑦 · 2))
51	19, 49, 50	sylancl 413	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 2 ∥ (𝑦 · 2))
52	51	ad2antrr 488	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 2 ∥ (𝑦 · 2))
53	52, 41	breqtrrd 4137	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 2 ∥ (𝑅‘𝑦))
54	53	adantr 276	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → 2 ∥ (𝑅‘𝑦))
55		eldifi 3341	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
56		prmz 12808	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
57	55, 56	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℤ)
58	35, 57	syl 14	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 𝑃 ∈ ℤ)
59		oddn2prm 12959	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → ¬ 2 ∥ 𝑃)
60	1, 59	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ¬ 2 ∥ 𝑃)
61	60	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → ¬ 2 ∥ 𝑃)
62	49	a1i 9	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 2 ∈ ℤ)
63	24, 62	zmulcld 9706	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → (𝑧 · 2) ∈ ℤ)
64		dvdsmul2 12500	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑧 ∈ ℤ ∧ 2 ∈ ℤ) → 2 ∥ (𝑧 · 2))
65	24, 49, 64	sylancl 413	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 2 ∥ (𝑧 · 2))
66		omeo 12584	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ ℤ ∧ ¬ 2 ∥ 𝑃) ∧ ((𝑧 · 2) ∈ ℤ ∧ 2 ∥ (𝑧 · 2))) → ¬ 2 ∥ (𝑃 − (𝑧 · 2)))
67	58, 61, 63, 65, 66	syl22anc 1275	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → ¬ 2 ∥ (𝑃 − (𝑧 · 2)))
68	67	adantr 276	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → ¬ 2 ∥ (𝑃 − (𝑧 · 2)))
69	47	adantr 276	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → (𝑅‘𝑧) = if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))))
70		simpr 110	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → ¬ (𝑧 · 2) < (𝑃 / 2))
71	70	iffalsed 3632	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))) = (𝑃 − (𝑧 · 2)))
72	69, 71	eqtrd 2265	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → (𝑅‘𝑧) = (𝑃 − (𝑧 · 2)))
73	72	breq2d 4121	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → (2 ∥ (𝑅‘𝑧) ↔ 2 ∥ (𝑃 − (𝑧 · 2))))
74	68, 73	mtbird 680	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → ¬ 2 ∥ (𝑅‘𝑧))
75	74	ad4ant14 514	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → ¬ 2 ∥ (𝑅‘𝑧))
76		simpr 110	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → (𝑅‘𝑦) = (𝑅‘𝑧))
77	76	breq2d 4121	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → (2 ∥ (𝑅‘𝑦) ↔ 2 ∥ (𝑅‘𝑧)))
78	77	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → (2 ∥ (𝑅‘𝑦) ↔ 2 ∥ (𝑅‘𝑧)))
79	75, 78	mtbird 680	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → ¬ 2 ∥ (𝑅‘𝑦))
80	54, 79	pm2.65da 667	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → ¬ ¬ (𝑧 · 2) < (𝑃 / 2))
81		zq 9958	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 · 2) ∈ ℤ → (𝑧 · 2) ∈ ℚ)
82	63, 81	syl 14	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → (𝑧 · 2) ∈ ℚ)
83	1, 57	syl 14	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑃 ∈ ℤ)
84		2nn 9399	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
85		znq 9956	. . . . . . . . . . . . . . . 16 ⊢ ((𝑃 ∈ ℤ ∧ 2 ∈ ℕ) → (𝑃 / 2) ∈ ℚ)
86	83, 84, 85	sylancl 413	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑃 / 2) ∈ ℚ)
87	86	adantr 276	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → (𝑃 / 2) ∈ ℚ)
88		qdclt 10605	. . . . . . . . . . . . . 14 ⊢ (((𝑧 · 2) ∈ ℚ ∧ (𝑃 / 2) ∈ ℚ) → DECID (𝑧 · 2) < (𝑃 / 2))
89	82, 87, 88	syl2anc 411	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → DECID (𝑧 · 2) < (𝑃 / 2))
90		exmiddc 844	. . . . . . . . . . . . 13 ⊢ (DECID (𝑧 · 2) < (𝑃 / 2) → ((𝑧 · 2) < (𝑃 / 2) ∨ ¬ (𝑧 · 2) < (𝑃 / 2)))
91	89, 90	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → ((𝑧 · 2) < (𝑃 / 2) ∨ ¬ (𝑧 · 2) < (𝑃 / 2)))
92	91	ad2antrr 488	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → ((𝑧 · 2) < (𝑃 / 2) ∨ ¬ (𝑧 · 2) < (𝑃 / 2)))
93	80, 92	ecased 1386	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑧 · 2) < (𝑃 / 2))
94	93	iftrued 3629	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))) = (𝑧 · 2))
95	48, 94	eqtrd 2265	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑧) = (𝑧 · 2))
96	30, 41, 95	3eqtr3d 2273	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑦 · 2) = (𝑧 · 2))
97	22, 26, 27, 29, 96	mulcanap2ad 8938	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 𝑦 = 𝑧)
98	19	zcnd 9701	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 𝑦 ∈ ℂ)
99	98	ad2antrr 488	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → 𝑦 ∈ ℂ)
100	24	zcnd 9701	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 𝑧 ∈ ℂ)
101	100	ad2antrr 488	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → 𝑧 ∈ ℂ)
102		2cnd 9310	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → 2 ∈ ℂ)
103	28	a1i 9	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → 2 # 0)
104	83	zcnd 9701	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℂ)
105	104	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → 𝑃 ∈ ℂ)
106	19, 62	zmulcld 9706	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → (𝑦 · 2) ∈ ℤ)
107	106	zcnd 9701	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → (𝑦 · 2) ∈ ℂ)
108	107	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → (𝑦 · 2) ∈ ℂ)
109	63	zcnd 9701	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → (𝑧 · 2) ∈ ℂ)
110	109	ad2antrr 488	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → (𝑧 · 2) ∈ ℂ)
111		simplr 529	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑦) = (𝑅‘𝑧))
112	37	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑦) = if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2))))
113		simpr 110	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → ¬ (𝑦 · 2) < (𝑃 / 2))
114	113	iffalsed 3632	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2))) = (𝑃 − (𝑦 · 2)))
115	112, 114	eqtrd 2265	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑦) = (𝑃 − (𝑦 · 2)))
116	47	ad2antrr 488	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑧) = if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))))
117	65	ad3antrrr 492	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → 2 ∥ (𝑧 · 2))
118	47	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → (𝑅‘𝑧) = if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))))
119		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → (𝑧 · 2) < (𝑃 / 2))
120	119	iftrued 3629	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))) = (𝑧 · 2))
121	118, 120	eqtrd 2265	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → (𝑅‘𝑧) = (𝑧 · 2))
122	117, 121	breqtrrd 4137	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → 2 ∥ (𝑅‘𝑧))
123	77	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → (2 ∥ (𝑅‘𝑦) ↔ 2 ∥ (𝑅‘𝑧)))
124	122, 123	mpbird 167	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → 2 ∥ (𝑅‘𝑦))
125		omeo 12584	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℤ ∧ ¬ 2 ∥ 𝑃) ∧ ((𝑦 · 2) ∈ ℤ ∧ 2 ∥ (𝑦 · 2))) → ¬ 2 ∥ (𝑃 − (𝑦 · 2)))
126	58, 61, 106, 51, 125	syl22anc 1275	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → ¬ 2 ∥ (𝑃 − (𝑦 · 2)))
127	126	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → ¬ 2 ∥ (𝑃 − (𝑦 · 2)))
128	37	ad3antrrr 492	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → (𝑅‘𝑦) = if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2))))
129		simplr 529	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → ¬ (𝑦 · 2) < (𝑃 / 2))
130	129	iffalsed 3632	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → if((𝑦 · 2) < (𝑃 / 2), (𝑦 · 2), (𝑃 − (𝑦 · 2))) = (𝑃 − (𝑦 · 2)))
131	128, 130	eqtrd 2265	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → (𝑅‘𝑦) = (𝑃 − (𝑦 · 2)))
132	131	breq2d 4121	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → (2 ∥ (𝑅‘𝑦) ↔ 2 ∥ (𝑃 − (𝑦 · 2))))
133	127, 132	mtbird 680	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) ∧ (𝑧 · 2) < (𝑃 / 2)) → ¬ 2 ∥ (𝑅‘𝑦))
134	124, 133	pm2.65da 667	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → ¬ (𝑧 · 2) < (𝑃 / 2))
135	134	iffalsed 3632	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → if((𝑧 · 2) < (𝑃 / 2), (𝑧 · 2), (𝑃 − (𝑧 · 2))) = (𝑃 − (𝑧 · 2)))
136	116, 135	eqtrd 2265	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑧) = (𝑃 − (𝑧 · 2)))
137	111, 115, 136	3eqtr3d 2273	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → (𝑃 − (𝑦 · 2)) = (𝑃 − (𝑧 · 2)))
138	105, 108, 110, 137	subcand 8625	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → (𝑦 · 2) = (𝑧 · 2))
139	99, 101, 102, 103, 138	mulcanap2ad 8938	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → 𝑦 = 𝑧)
140	49	a1i 9	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → 2 ∈ ℤ)
141	20, 140	zmulcld 9706	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → (𝑦 · 2) ∈ ℤ)
142		zq 9958	. . . . . . . . 9 ⊢ ((𝑦 · 2) ∈ ℤ → (𝑦 · 2) ∈ ℚ)
143	141, 142	syl 14	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → (𝑦 · 2) ∈ ℚ)
144	86	ad2antrr 488	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → (𝑃 / 2) ∈ ℚ)
145		qdclt 10605	. . . . . . . 8 ⊢ (((𝑦 · 2) ∈ ℚ ∧ (𝑃 / 2) ∈ ℚ) → DECID (𝑦 · 2) < (𝑃 / 2))
146	143, 144, 145	syl2anc 411	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → DECID (𝑦 · 2) < (𝑃 / 2))
147		exmiddc 844	. . . . . . 7 ⊢ (DECID (𝑦 · 2) < (𝑃 / 2) → ((𝑦 · 2) < (𝑃 / 2) ∨ ¬ (𝑦 · 2) < (𝑃 / 2)))
148	146, 147	syl 14	. . . . . 6 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → ((𝑦 · 2) < (𝑃 / 2) ∨ ¬ (𝑦 · 2) < (𝑃 / 2)))
149	97, 139, 148	mpjaodan 806	. . . . 5 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → 𝑦 = 𝑧)
150	149	ex 115	. . . 4 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → ((𝑅‘𝑦) = (𝑅‘𝑧) → 𝑦 = 𝑧))
151	150	ralrimivva 2624	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (1...𝐻)∀𝑧 ∈ (1...𝐻)((𝑅‘𝑦) = (𝑅‘𝑧) → 𝑦 = 𝑧))
152		dff13 5941	. . 3 ⊢ (𝑅:(1...𝐻)–1-1→(1...𝐻) ↔ (𝑅:(1...𝐻)⟶(1...𝐻) ∧ ∀𝑦 ∈ (1...𝐻)∀𝑧 ∈ (1...𝐻)((𝑅‘𝑦) = (𝑅‘𝑧) → 𝑦 = 𝑧)))
153	17, 151, 152	sylanbrc 417	. 2 ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1→(1...𝐻))
154		df-f1o 5359	. 2 ⊢ (𝑅:(1...𝐻)–1-1-onto→(1...𝐻) ↔ (𝑅:(1...𝐻)–1-1→(1...𝐻) ∧ 𝑅:(1...𝐻)–onto→(1...𝐻)))
155	153, 15, 154	sylanbrc 417	1 ⊢ (𝜑 → 𝑅:(1...𝐻)–1-1-onto→(1...𝐻))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 716 DECID wdc 842 = wceq 1398 ∈ wcel 2203 ∀wral 2520 ∖ cdif 3208 ifcif 3620 {csn 3689 class class class wbr 4109 ↦ cmpt 4171 ran crn 4750 Fn wfn 5347 ⟶wf 5348 –1-1→wf1 5349 –onto→wfo 5350 –1-1-onto→wf1o 5351 ‘cfv 5352 (class class class)co 6050 ℂcc 8125 0cc0 8127 1c1 8128 · cmul 8132 < clt 8308 − cmin 8444 # cap 8855 / cdiv 8946 ℕcn 9237 2c2 9288 ℤcz 9577 ℚcq 9951 ...cfz 10342 ∥ cdvds 12473 ℙcprime 12804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-coll 4225 ax-sep 4228 ax-nul 4236 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-iinf 4710 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 ax-pre-mulext 8245 ax-arch 8246 ax-caucvg 8247

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-xor 1421 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rmo 2528 df-rab 2529 df-v 2815 df-sbc 3043 df-csb 3139 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-nul 3509 df-if 3621 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-int 3950 df-iun 3993 df-br 4110 df-opab 4172 df-mpt 4173 df-tr 4209 df-id 4414 df-po 4417 df-iso 4418 df-iord 4487 df-on 4489 df-ilim 4490 df-suc 4492 df-iom 4713 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-res 4761 df-ima 4762 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-f1 5357 df-fo 5358 df-f1o 5359 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-1st 6334 df-2nd 6335 df-recs 6536 df-frec 6622 df-1o 6647 df-2o 6648 df-er 6767 df-en 6976 df-pnf 8310 df-mnf 8311 df-xr 8312 df-ltxr 8313 df-le 8314 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 df-div 8947 df-inn 9238 df-2 9296 df-3 9297 df-4 9298 df-n0 9497 df-z 9578 df-uz 9854 df-q 9952 df-rp 9987 df-ioo 10225 df-fz 10343 df-fl 10630 df-mod 10685 df-seqfrec 10810 df-exp 10901 df-cj 11527 df-re 11528 df-im 11529 df-rsqrt 11683 df-abs 11684 df-dvds 12474 df-prm 12805

This theorem is referenced by: gausslemma2dlem1 15934

W3C validator