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Theorem gausslemma2dlem1f1o 15118

Description: Lemma for gausslemma2dlem1 15119. (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 15117	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝐻)) → if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))) ∈ ℤ)
7	6	ralrimiva 2567	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (1...𝐻)if((𝑥 · 2) < (𝑃 / 2), (𝑥 · 2), (𝑃 − (𝑥 · 2))) ∈ ℤ)
8	4	fnmpt 5372	. . . . . . 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 5474	. . . . . 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 15116	. . . . . 6 ⊢ (𝜑 → ran 𝑅 = (1...𝐻))
13		foeq3 5466	. . . . . 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 5468	. . . 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 529	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 𝑦 ∈ (1...𝐻))
19	18	elfzelzd 10082	. . . . . . . . . 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 9430	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 𝑦 ∈ ℂ)
23		simprr 531	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → 𝑧 ∈ (1...𝐻))
24	23	elfzelzd 10082	. . . . . . . . 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 9430	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 𝑧 ∈ ℂ)
27		2cnd 9045	. . . . . . 7 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 2 ∈ ℂ)
28		2ap0 9065	. . . . . . . 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 528	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑦) = (𝑅‘𝑧))
31		oveq1 5917	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥 · 2) = (𝑦 · 2))
32	31	breq1d 4039	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑦 · 2) < (𝑃 / 2)))
33	31	oveq2d 5926	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑦 · 2)))
34	32, 31, 33	ifbieq12d 3583	. . . . . . . . . . 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 15117	. . . . . . . . . . 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 5643	. . . . . . . . . 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 3564	. . . . . . . . 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 2226	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑅‘𝑦) = (𝑦 · 2))
42		oveq1 5917	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥 · 2) = (𝑧 · 2))
43	42	breq1d 4039	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → ((𝑥 · 2) < (𝑃 / 2) ↔ (𝑧 · 2) < (𝑃 / 2)))
44	42	oveq2d 5926	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑃 − (𝑥 · 2)) = (𝑃 − (𝑧 · 2)))
45	43, 42, 44	ifbieq12d 3583	. . . . . . . . . . 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 15117	. . . . . . . . . . 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 5643	. . . . . . . . . 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 9335	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℤ
50		dvdsmul2 11944	. . . . . . . . . . . . . . . 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 4057	. . . . . . . . . . . . 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 3281	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑃 ∈ (ℙ ∖ {2}) → 𝑃 ∈ ℙ)
56		prmz 12236	. . . . . . . . . . . . . . . . . . 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 12386	. . . . . . . . . . . . . . . . . . 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 9435	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → (𝑧 · 2) ∈ ℤ)
64		dvdsmul2 11944	. . . . . . . . . . . . . . . . . 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 12026	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑃 ∈ ℤ ∧ ¬ 2 ∥ 𝑃) ∧ ((𝑧 · 2) ∈ ℤ ∧ 2 ∥ (𝑧 · 2))) → ¬ 2 ∥ (𝑃 − (𝑧 · 2)))
67	58, 61, 63, 65, 66	syl22anc 1250	. . . . . . . . . . . . . . . 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 3567	. . . . . . . . . . . . . . . . 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 2226	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → (𝑅‘𝑧) = (𝑃 − (𝑧 · 2)))
73	72	breq2d 4041	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ ¬ (𝑧 · 2) < (𝑃 / 2)) → (2 ∥ (𝑅‘𝑧) ↔ 2 ∥ (𝑃 − (𝑧 · 2))))
74	68, 73	mtbird 674	. . . . . . . . . . . . . 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 4041	. . . . . . . . . . . . . 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 674	. . . . . . . . . . . 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 662	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → ¬ ¬ (𝑧 · 2) < (𝑃 / 2))
81		zq 9681	. . . . . . . . . . . . . . 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 9133	. . . . . . . . . . . . . . . 16 ⊢ 2 ∈ ℕ
85		znq 9679	. . . . . . . . . . . . . . . 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 10305	. . . . . . . . . . . . . 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 837	. . . . . . . . . . . . 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 1360	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → (𝑧 · 2) < (𝑃 / 2))
94	93	iftrued 3564	. . . . . . . . 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 2226	. . . . . . . 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 2234	. . . . . . 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 8673	. . . . . 6 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ (𝑦 · 2) < (𝑃 / 2)) → 𝑦 = 𝑧)
98	19	zcnd 9430	. . . . . . . 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 9430	. . . . . . . 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 9045	. . . . . . 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 9430	. . . . . . . . 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 9435	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) → (𝑦 · 2) ∈ ℤ)
107	106	zcnd 9430	. . . . . . . . 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 9430	. . . . . . . . 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 528	. . . . . . . . 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 3567	. . . . . . . . . 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 2226	. . . . . . . . 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 3564	. . . . . . . . . . . . . . 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 2226	. . . . . . . . . . . . . 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 4057	. . . . . . . . . . . . 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 12026	. . . . . . . . . . . . . . 15 ⊢ (((𝑃 ∈ ℤ ∧ ¬ 2 ∥ 𝑃) ∧ ((𝑦 · 2) ∈ ℤ ∧ 2 ∥ (𝑦 · 2))) → ¬ 2 ∥ (𝑃 − (𝑦 · 2)))
126	58, 61, 106, 51, 125	syl22anc 1250	. . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . . . 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 3567	. . . . . . . . . . . . . . 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 2226	. . . . . . . . . . . . . 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 4041	. . . . . . . . . . . . 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 674	. . . . . . . . . . . 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 662	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) ∧ ¬ (𝑦 · 2) < (𝑃 / 2)) → ¬ (𝑧 · 2) < (𝑃 / 2))
135	134	iffalsed 3567	. . . . . . . . . 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 2226	. . . . . . . . 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 2234	. . . . . . . 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 8361	. . . . . . 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 8673	. . . . . 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 9435	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑦 ∈ (1...𝐻) ∧ 𝑧 ∈ (1...𝐻))) ∧ (𝑅‘𝑦) = (𝑅‘𝑧)) → (𝑦 · 2) ∈ ℤ)
142		zq 9681	. . . . . . . . 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 10305	. . . . . . . 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 837	. . . . . . 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 799	. . . . 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 2576	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ (1...𝐻)∀𝑧 ∈ (1...𝐻)((𝑅‘𝑦) = (𝑅‘𝑧) → 𝑦 = 𝑧))
152		dff13 5803	. . 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 5253	. 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 709 DECID wdc 835 = wceq 1364 ∈ wcel 2164 ∀wral 2472 ∖ cdif 3150 ifcif 3557 {csn 3618 class class class wbr 4029 ↦ cmpt 4090 ran crn 4656 Fn wfn 5241 ⟶wf 5242 –1-1→wf1 5243 –onto→wfo 5244 –1-1-onto→wf1o 5245 ‘cfv 5246 (class class class)co 5910 ℂcc 7860 0cc0 7862 1c1 7863 · cmul 7867 < clt 8044 − cmin 8180 # cap 8590 / cdiv 8681 ℕcn 8972 2c2 9023 ℤcz 9307 ℚcq 9674 ...cfz 10064 ∥ cdvds 11917 ℙcprime 12232

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4462 ax-setind 4565 ax-iinf 4616 ax-cnex 7953 ax-resscn 7954 ax-1cn 7955 ax-1re 7956 ax-icn 7957 ax-addcl 7958 ax-addrcl 7959 ax-mulcl 7960 ax-mulrcl 7961 ax-addcom 7962 ax-mulcom 7963 ax-addass 7964 ax-mulass 7965 ax-distr 7966 ax-i2m1 7967 ax-0lt1 7968 ax-1rid 7969 ax-0id 7970 ax-rnegex 7971 ax-precex 7972 ax-cnre 7973 ax-pre-ltirr 7974 ax-pre-ltwlin 7975 ax-pre-lttrn 7976 ax-pre-apti 7977 ax-pre-ltadd 7978 ax-pre-mulgt0 7979 ax-pre-mulext 7980 ax-arch 7981 ax-caucvg 7982

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-xor 1387 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4322 df-po 4325 df-iso 4326 df-iord 4395 df-on 4397 df-ilim 4398 df-suc 4400 df-iom 4619 df-xp 4661 df-rel 4662 df-cnv 4663 df-co 4664 df-dm 4665 df-rn 4666 df-res 4667 df-ima 4668 df-iota 5207 df-fun 5248 df-fn 5249 df-f 5250 df-f1 5251 df-fo 5252 df-f1o 5253 df-fv 5254 df-riota 5865 df-ov 5913 df-oprab 5914 df-mpo 5915 df-1st 6184 df-2nd 6185 df-recs 6349 df-frec 6435 df-1o 6460 df-2o 6461 df-er 6578 df-en 6786 df-pnf 8046 df-mnf 8047 df-xr 8048 df-ltxr 8049 df-le 8050 df-sub 8182 df-neg 8183 df-reap 8584 df-ap 8591 df-div 8682 df-inn 8973 df-2 9031 df-3 9032 df-4 9033 df-n0 9231 df-z 9308 df-uz 9583 df-q 9675 df-rp 9710 df-ioo 9948 df-fz 10065 df-fl 10329 df-mod 10384 df-seqfrec 10509 df-exp 10597 df-cj 10973 df-re 10974 df-im 10975 df-rsqrt 11129 df-abs 11130 df-dvds 11918 df-prm 12233

This theorem is referenced by: gausslemma2dlem1 15119

W3C validator