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Theorem 4sqlem11 16827
Description: Lemma for 4sq 16836. Use the pigeonhole principle to show that the sets {𝑚↑2 ∣ 𝑚 ∈ (0...𝑁)} and {-1 − 𝑛↑2 ∣ 𝑛 ∈ (0...𝑁)} have a common element, mod 𝑃. (Contributed by Mario Carneiro, 15-Jul-2014.)
Hypotheses
Ref Expression
4sq.1 𝑆 = {𝑛 ∣ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℤ ∃𝑧 ∈ ℤ ∃𝑤 ∈ ℤ 𝑛 = (((𝑥↑2) + (𝑦↑2)) + ((𝑧↑2) + (𝑤↑2)))}
4sq.2 (𝜑𝑁 ∈ ℕ)
4sq.3 (𝜑𝑃 = ((2 · 𝑁) + 1))
4sq.4 (𝜑𝑃 ∈ ℙ)
4sqlem11.5 𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
4sqlem11.6 𝐹 = (𝑣𝐴 ↦ ((𝑃 − 1) − 𝑣))
Assertion
Ref Expression
4sqlem11 (𝜑 → (𝐴 ∩ ran 𝐹) ≠ ∅)
Distinct variable groups:   𝑤,𝑛,𝑥,𝑦,𝑧   𝑣,𝑛,𝐴   𝑛,𝐹   𝑢,𝑛,𝑚,𝑣,𝑁   𝑃,𝑚,𝑛,𝑢,𝑣   𝜑,𝑚,𝑛,𝑢,𝑣   𝑆,𝑚,𝑛,𝑢,𝑣
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝐴(𝑥,𝑦,𝑧,𝑤,𝑢,𝑚)   𝑃(𝑥,𝑦,𝑧,𝑤)   𝑆(𝑥,𝑦,𝑧,𝑤)   𝐹(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑚)   𝑁(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 4sqlem11
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fzfid 13878 . . . . . 6 (𝜑 → (0...(𝑃 − 1)) ∈ Fin)
2 4sqlem11.5 . . . . . . . 8 𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
3 elfzelz 13441 . . . . . . . . . . . . 13 (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
4 zsqcl 14034 . . . . . . . . . . . . 13 (𝑚 ∈ ℤ → (𝑚↑2) ∈ ℤ)
53, 4syl 17 . . . . . . . . . . . 12 (𝑚 ∈ (0...𝑁) → (𝑚↑2) ∈ ℤ)
6 4sq.4 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ ℙ)
7 prmnn 16550 . . . . . . . . . . . . 13 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
86, 7syl 17 . . . . . . . . . . . 12 (𝜑𝑃 ∈ ℕ)
9 zmodfz 13798 . . . . . . . . . . . 12 (((𝑚↑2) ∈ ℤ ∧ 𝑃 ∈ ℕ) → ((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1)))
105, 8, 9syl2anr 597 . . . . . . . . . . 11 ((𝜑𝑚 ∈ (0...𝑁)) → ((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1)))
11 eleq1a 2833 . . . . . . . . . . 11 (((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1)) → (𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ (0...(𝑃 − 1))))
1210, 11syl 17 . . . . . . . . . 10 ((𝜑𝑚 ∈ (0...𝑁)) → (𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ (0...(𝑃 − 1))))
1312rexlimdva 3152 . . . . . . . . 9 (𝜑 → (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ (0...(𝑃 − 1))))
1413abssdv 4025 . . . . . . . 8 (𝜑 → {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} ⊆ (0...(𝑃 − 1)))
152, 14eqsstrid 3992 . . . . . . 7 (𝜑𝐴 ⊆ (0...(𝑃 − 1)))
16 prmz 16551 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
176, 16syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑃 ∈ ℤ)
18 peano2zm 12546 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
1917, 18syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑃 − 1) ∈ ℤ)
2019zcnd 12608 . . . . . . . . . . . . 13 (𝜑 → (𝑃 − 1) ∈ ℂ)
2120addid2d 11356 . . . . . . . . . . . 12 (𝜑 → (0 + (𝑃 − 1)) = (𝑃 − 1))
2221oveq1d 7372 . . . . . . . . . . 11 (𝜑 → ((0 + (𝑃 − 1)) − 𝑣) = ((𝑃 − 1) − 𝑣))
2322adantr 481 . . . . . . . . . 10 ((𝜑𝑣𝐴) → ((0 + (𝑃 − 1)) − 𝑣) = ((𝑃 − 1) − 𝑣))
2415sselda 3944 . . . . . . . . . . 11 ((𝜑𝑣𝐴) → 𝑣 ∈ (0...(𝑃 − 1)))
25 fzrev3i 13508 . . . . . . . . . . 11 (𝑣 ∈ (0...(𝑃 − 1)) → ((0 + (𝑃 − 1)) − 𝑣) ∈ (0...(𝑃 − 1)))
2624, 25syl 17 . . . . . . . . . 10 ((𝜑𝑣𝐴) → ((0 + (𝑃 − 1)) − 𝑣) ∈ (0...(𝑃 − 1)))
2723, 26eqeltrrd 2839 . . . . . . . . 9 ((𝜑𝑣𝐴) → ((𝑃 − 1) − 𝑣) ∈ (0...(𝑃 − 1)))
28 4sqlem11.6 . . . . . . . . 9 𝐹 = (𝑣𝐴 ↦ ((𝑃 − 1) − 𝑣))
2927, 28fmptd 7062 . . . . . . . 8 (𝜑𝐹:𝐴⟶(0...(𝑃 − 1)))
3029frnd 6676 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ (0...(𝑃 − 1)))
3115, 30unssd 4146 . . . . . 6 (𝜑 → (𝐴 ∪ ran 𝐹) ⊆ (0...(𝑃 − 1)))
321, 31ssfid 9211 . . . . 5 (𝜑 → (𝐴 ∪ ran 𝐹) ∈ Fin)
33 hashcl 14256 . . . . 5 ((𝐴 ∪ ran 𝐹) ∈ Fin → (♯‘(𝐴 ∪ ran 𝐹)) ∈ ℕ0)
3432, 33syl 17 . . . 4 (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ∈ ℕ0)
3534nn0red 12474 . . 3 (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ∈ ℝ)
3617zred 12607 . . 3 (𝜑𝑃 ∈ ℝ)
37 ssdomg 8940 . . . . . 6 ((0...(𝑃 − 1)) ∈ Fin → ((𝐴 ∪ ran 𝐹) ⊆ (0...(𝑃 − 1)) → (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1))))
381, 31, 37sylc 65 . . . . 5 (𝜑 → (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1)))
39 hashdom 14279 . . . . . 6 (((𝐴 ∪ ran 𝐹) ∈ Fin ∧ (0...(𝑃 − 1)) ∈ Fin) → ((♯‘(𝐴 ∪ ran 𝐹)) ≤ (♯‘(0...(𝑃 − 1))) ↔ (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1))))
4032, 1, 39syl2anc 584 . . . . 5 (𝜑 → ((♯‘(𝐴 ∪ ran 𝐹)) ≤ (♯‘(0...(𝑃 − 1))) ↔ (𝐴 ∪ ran 𝐹) ≼ (0...(𝑃 − 1))))
4138, 40mpbird 256 . . . 4 (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ≤ (♯‘(0...(𝑃 − 1))))
42 fz01en 13469 . . . . . . 7 (𝑃 ∈ ℤ → (0...(𝑃 − 1)) ≈ (1...𝑃))
4317, 42syl 17 . . . . . 6 (𝜑 → (0...(𝑃 − 1)) ≈ (1...𝑃))
44 fzfid 13878 . . . . . . 7 (𝜑 → (1...𝑃) ∈ Fin)
45 hashen 14247 . . . . . . 7 (((0...(𝑃 − 1)) ∈ Fin ∧ (1...𝑃) ∈ Fin) → ((♯‘(0...(𝑃 − 1))) = (♯‘(1...𝑃)) ↔ (0...(𝑃 − 1)) ≈ (1...𝑃)))
461, 44, 45syl2anc 584 . . . . . 6 (𝜑 → ((♯‘(0...(𝑃 − 1))) = (♯‘(1...𝑃)) ↔ (0...(𝑃 − 1)) ≈ (1...𝑃)))
4743, 46mpbird 256 . . . . 5 (𝜑 → (♯‘(0...(𝑃 − 1))) = (♯‘(1...𝑃)))
488nnnn0d 12473 . . . . . 6 (𝜑𝑃 ∈ ℕ0)
49 hashfz1 14246 . . . . . 6 (𝑃 ∈ ℕ0 → (♯‘(1...𝑃)) = 𝑃)
5048, 49syl 17 . . . . 5 (𝜑 → (♯‘(1...𝑃)) = 𝑃)
5147, 50eqtrd 2776 . . . 4 (𝜑 → (♯‘(0...(𝑃 − 1))) = 𝑃)
5241, 51breqtrd 5131 . . 3 (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ≤ 𝑃)
5335, 36, 52lensymd 11306 . 2 (𝜑 → ¬ 𝑃 < (♯‘(𝐴 ∪ ran 𝐹)))
5436adantr 481 . . . . . 6 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 ∈ ℝ)
5554ltp1d 12085 . . . . 5 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 < (𝑃 + 1))
56 4sq.2 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ)
5756nncnd 12169 . . . . . . . . 9 (𝜑𝑁 ∈ ℂ)
58 1cnd 11150 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
5957, 57, 58, 58add4d 11383 . . . . . . . 8 (𝜑 → ((𝑁 + 𝑁) + (1 + 1)) = ((𝑁 + 1) + (𝑁 + 1)))
60 4sq.3 . . . . . . . . . 10 (𝜑𝑃 = ((2 · 𝑁) + 1))
6160oveq1d 7372 . . . . . . . . 9 (𝜑 → (𝑃 + 1) = (((2 · 𝑁) + 1) + 1))
62 2cn 12228 . . . . . . . . . . 11 2 ∈ ℂ
63 mulcl 11135 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (2 · 𝑁) ∈ ℂ)
6462, 57, 63sylancr 587 . . . . . . . . . 10 (𝜑 → (2 · 𝑁) ∈ ℂ)
6564, 58, 58addassd 11177 . . . . . . . . 9 (𝜑 → (((2 · 𝑁) + 1) + 1) = ((2 · 𝑁) + (1 + 1)))
66572timesd 12396 . . . . . . . . . 10 (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
6766oveq1d 7372 . . . . . . . . 9 (𝜑 → ((2 · 𝑁) + (1 + 1)) = ((𝑁 + 𝑁) + (1 + 1)))
6861, 65, 673eqtrd 2780 . . . . . . . 8 (𝜑 → (𝑃 + 1) = ((𝑁 + 𝑁) + (1 + 1)))
6910ex 413 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑚 ∈ (0...𝑁) → ((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1))))
708adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℕ)
713ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℤ)
7271, 4syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚↑2) ∈ ℤ)
73 elfzelz 13441 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 ∈ (0...𝑁) → 𝑢 ∈ ℤ)
7473ad2antll 727 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℤ)
75 zsqcl 14034 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 ∈ ℤ → (𝑢↑2) ∈ ℤ)
7674, 75syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑢↑2) ∈ ℤ)
77 moddvds 16147 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃 ∈ ℕ ∧ (𝑚↑2) ∈ ℤ ∧ (𝑢↑2) ∈ ℤ) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑃 ∥ ((𝑚↑2) − (𝑢↑2))))
7870, 72, 76, 77syl3anc 1371 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑃 ∥ ((𝑚↑2) − (𝑢↑2))))
7971zcnd 12608 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℂ)
8074zcnd 12608 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℂ)
81 subsq 14114 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑚 ∈ ℂ ∧ 𝑢 ∈ ℂ) → ((𝑚↑2) − (𝑢↑2)) = ((𝑚 + 𝑢) · (𝑚𝑢)))
8279, 80, 81syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚↑2) − (𝑢↑2)) = ((𝑚 + 𝑢) · (𝑚𝑢)))
8382breq2d 5117 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ ((𝑚↑2) − (𝑢↑2)) ↔ 𝑃 ∥ ((𝑚 + 𝑢) · (𝑚𝑢))))
846adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℙ)
8571, 74zaddcld 12611 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ∈ ℤ)
8671, 74zsubcld 12612 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚𝑢) ∈ ℤ)
87 euclemma 16589 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃 ∈ ℙ ∧ (𝑚 + 𝑢) ∈ ℤ ∧ (𝑚𝑢) ∈ ℤ) → (𝑃 ∥ ((𝑚 + 𝑢) · (𝑚𝑢)) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚𝑢))))
8884, 85, 86, 87syl3anc 1371 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ ((𝑚 + 𝑢) · (𝑚𝑢)) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚𝑢))))
8978, 83, 883bitrd 304 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚𝑢))))
9085zred 12607 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ∈ ℝ)
91 2re 12227 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 ∈ ℝ
9256nnred 12168 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑁 ∈ ℝ)
93 remulcl 11136 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (2 · 𝑁) ∈ ℝ)
9491, 92, 93sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (2 · 𝑁) ∈ ℝ)
9594adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) ∈ ℝ)
9684, 16syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℤ)
9796zred 12607 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℝ)
9871zred 12607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
9974zred 12607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℝ)
10092adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
101 elfzle2 13445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑚 ∈ (0...𝑁) → 𝑚𝑁)
102101ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚𝑁)
103 elfzle2 13445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑢 ∈ (0...𝑁) → 𝑢𝑁)
104103ad2antll 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢𝑁)
10598, 99, 100, 100, 102, 104le2addd 11774 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ≤ (𝑁 + 𝑁))
10657adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑁 ∈ ℂ)
1071062timesd 12396 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) = (𝑁 + 𝑁))
108105, 107breqtrrd 5133 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ≤ (2 · 𝑁))
10994ltp1d 12085 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (2 · 𝑁) < ((2 · 𝑁) + 1))
110109, 60breqtrrd 5133 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (2 · 𝑁) < 𝑃)
111110adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) < 𝑃)
11290, 95, 97, 108, 111lelttrd 11313 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) < 𝑃)
11390, 97ltnled 11302 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚 + 𝑢) < 𝑃 ↔ ¬ 𝑃 ≤ (𝑚 + 𝑢)))
114112, 113mpbid 231 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ¬ 𝑃 ≤ (𝑚 + 𝑢))
115114adantr 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → ¬ 𝑃 ≤ (𝑚 + 𝑢))
11617ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 𝑃 ∈ ℤ)
11785adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑚 + 𝑢) ∈ ℤ)
118 1red 11156 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 1 ∈ ℝ)
119 nn0abscl 15197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑚𝑢) ∈ ℤ → (abs‘(𝑚𝑢)) ∈ ℕ0)
12086, 119syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚𝑢)) ∈ ℕ0)
121120nn0red 12474 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚𝑢)) ∈ ℝ)
122121adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ∈ ℝ)
123117zred 12607 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑚 + 𝑢) ∈ ℝ)
124120adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ∈ ℕ0)
125124nn0zd 12525 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ∈ ℤ)
12686zcnd 12608 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚𝑢) ∈ ℂ)
127126adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑚𝑢) ∈ ℂ)
12879, 80subeq0ad 11522 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚𝑢) = 0 ↔ 𝑚 = 𝑢))
129128necon3bid 2988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚𝑢) ≠ 0 ↔ 𝑚𝑢))
130129biimpar 478 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑚𝑢) ≠ 0)
131127, 130absrpcld 15333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ∈ ℝ+)
132131rpgt0d 12960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 0 < (abs‘(𝑚𝑢)))
133 elnnz 12509 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((abs‘(𝑚𝑢)) ∈ ℕ ↔ ((abs‘(𝑚𝑢)) ∈ ℤ ∧ 0 < (abs‘(𝑚𝑢))))
134125, 132, 133sylanbrc 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ∈ ℕ)
135134nnge1d 12201 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 1 ≤ (abs‘(𝑚𝑢)))
136 0cnd 11148 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ∈ ℂ)
13779, 80, 136abs3difd 15345 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚𝑢)) ≤ ((abs‘(𝑚 − 0)) + (abs‘(0 − 𝑢))))
13879subid1d 11501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 − 0) = 𝑚)
139138fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 0)) = (abs‘𝑚))
140 elfzle1 13444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 ∈ (0...𝑁) → 0 ≤ 𝑚)
141140ad2antrl 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ≤ 𝑚)
14298, 141absidd 15307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘𝑚) = 𝑚)
143139, 142eqtrd 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 0)) = 𝑚)
144 0cn 11147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 0 ∈ ℂ
145 abssub 15211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((0 ∈ ℂ ∧ 𝑢 ∈ ℂ) → (abs‘(0 − 𝑢)) = (abs‘(𝑢 − 0)))
146144, 80, 145sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(0 − 𝑢)) = (abs‘(𝑢 − 0)))
14780subid1d 11501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑢 − 0) = 𝑢)
148147fveq2d 6846 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑢 − 0)) = (abs‘𝑢))
149 elfzle1 13444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑢 ∈ (0...𝑁) → 0 ≤ 𝑢)
150149ad2antll 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ≤ 𝑢)
15199, 150absidd 15307 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘𝑢) = 𝑢)
152146, 148, 1513eqtrd 2780 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(0 − 𝑢)) = 𝑢)
153143, 152oveq12d 7375 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((abs‘(𝑚 − 0)) + (abs‘(0 − 𝑢))) = (𝑚 + 𝑢))
154137, 153breqtrd 5131 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚𝑢)) ≤ (𝑚 + 𝑢))
155154adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ≤ (𝑚 + 𝑢))
156118, 122, 123, 135, 155letrd 11312 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 1 ≤ (𝑚 + 𝑢))
157 elnnz1 12529 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑚 + 𝑢) ∈ ℕ ↔ ((𝑚 + 𝑢) ∈ ℤ ∧ 1 ≤ (𝑚 + 𝑢)))
158117, 156, 157sylanbrc 583 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑚 + 𝑢) ∈ ℕ)
159 dvdsle 16192 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑃 ∈ ℤ ∧ (𝑚 + 𝑢) ∈ ℕ) → (𝑃 ∥ (𝑚 + 𝑢) → 𝑃 ≤ (𝑚 + 𝑢)))
160116, 158, 159syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑃 ∥ (𝑚 + 𝑢) → 𝑃 ≤ (𝑚 + 𝑢)))
161115, 160mtod 197 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → ¬ 𝑃 ∥ (𝑚 + 𝑢))
162161ex 413 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚𝑢 → ¬ 𝑃 ∥ (𝑚 + 𝑢)))
163162necon4ad 2962 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚 + 𝑢) → 𝑚 = 𝑢))
164 dvdsabsb 16158 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑃 ∈ ℤ ∧ (𝑚𝑢) ∈ ℤ) → (𝑃 ∥ (𝑚𝑢) ↔ 𝑃 ∥ (abs‘(𝑚𝑢))))
16596, 86, 164syl2anc 584 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚𝑢) ↔ 𝑃 ∥ (abs‘(𝑚𝑢))))
166 letr 11249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃 ∈ ℝ ∧ (abs‘(𝑚𝑢)) ∈ ℝ ∧ (𝑚 + 𝑢) ∈ ℝ) → ((𝑃 ≤ (abs‘(𝑚𝑢)) ∧ (abs‘(𝑚𝑢)) ≤ (𝑚 + 𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
16797, 121, 90, 166syl3anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑃 ≤ (abs‘(𝑚𝑢)) ∧ (abs‘(𝑚𝑢)) ≤ (𝑚 + 𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
168154, 167mpan2d 692 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ≤ (abs‘(𝑚𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
169114, 168mtod 197 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ¬ 𝑃 ≤ (abs‘(𝑚𝑢)))
170169adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → ¬ 𝑃 ≤ (abs‘(𝑚𝑢)))
17196adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 𝑃 ∈ ℤ)
172 dvdsle 16192 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑃 ∈ ℤ ∧ (abs‘(𝑚𝑢)) ∈ ℕ) → (𝑃 ∥ (abs‘(𝑚𝑢)) → 𝑃 ≤ (abs‘(𝑚𝑢))))
173171, 134, 172syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑃 ∥ (abs‘(𝑚𝑢)) → 𝑃 ≤ (abs‘(𝑚𝑢))))
174170, 173mtod 197 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → ¬ 𝑃 ∥ (abs‘(𝑚𝑢)))
175174ex 413 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚𝑢 → ¬ 𝑃 ∥ (abs‘(𝑚𝑢))))
176175necon4ad 2962 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (abs‘(𝑚𝑢)) → 𝑚 = 𝑢))
177165, 176sylbid 239 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚𝑢) → 𝑚 = 𝑢))
178163, 177jaod 857 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚𝑢)) → 𝑚 = 𝑢))
17989, 178sylbid 239 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) → 𝑚 = 𝑢))
180 oveq1 7364 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑢 → (𝑚↑2) = (𝑢↑2))
181180oveq1d 7372 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑢 → ((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃))
182179, 181impbid1 224 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑚 = 𝑢))
183182ex 413 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁)) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑚 = 𝑢)))
18469, 183dom2lem 8932 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1→(0...(𝑃 − 1)))
185 f1f1orn 6795 . . . . . . . . . . . . . . . 16 ((𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1→(0...(𝑃 − 1)) → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)))
186184, 185syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)))
187 eqid 2736 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)) = (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃))
188187rnmpt 5910 . . . . . . . . . . . . . . . . 17 ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)) = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
1892, 188eqtr4i 2767 . . . . . . . . . . . . . . . 16 𝐴 = ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃))
190 f1oeq3 6774 . . . . . . . . . . . . . . . 16 (𝐴 = ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)) → ((𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto𝐴 ↔ (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃))))
191189, 190ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto𝐴 ↔ (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto→ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)))
192186, 191sylibr 233 . . . . . . . . . . . . . 14 (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto𝐴)
193 ovex 7390 . . . . . . . . . . . . . . 15 (0...𝑁) ∈ V
194193f1oen 8913 . . . . . . . . . . . . . 14 ((𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto𝐴 → (0...𝑁) ≈ 𝐴)
195192, 194syl 17 . . . . . . . . . . . . 13 (𝜑 → (0...𝑁) ≈ 𝐴)
196195ensymd 8945 . . . . . . . . . . . 12 (𝜑𝐴 ≈ (0...𝑁))
197 ax-1cn 11109 . . . . . . . . . . . . . . 15 1 ∈ ℂ
198 pncan 11407 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
19957, 197, 198sylancl 586 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
200199oveq2d 7373 . . . . . . . . . . . . 13 (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
20156nnnn0d 12473 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℕ0)
202 peano2nn0 12453 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
203201, 202syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 + 1) ∈ ℕ0)
204203nn0zd 12525 . . . . . . . . . . . . . 14 (𝜑 → (𝑁 + 1) ∈ ℤ)
205 fz01en 13469 . . . . . . . . . . . . . 14 ((𝑁 + 1) ∈ ℤ → (0...((𝑁 + 1) − 1)) ≈ (1...(𝑁 + 1)))
206204, 205syl 17 . . . . . . . . . . . . 13 (𝜑 → (0...((𝑁 + 1) − 1)) ≈ (1...(𝑁 + 1)))
207200, 206eqbrtrrd 5129 . . . . . . . . . . . 12 (𝜑 → (0...𝑁) ≈ (1...(𝑁 + 1)))
208 entr 8946 . . . . . . . . . . . 12 ((𝐴 ≈ (0...𝑁) ∧ (0...𝑁) ≈ (1...(𝑁 + 1))) → 𝐴 ≈ (1...(𝑁 + 1)))
209196, 207, 208syl2anc 584 . . . . . . . . . . 11 (𝜑𝐴 ≈ (1...(𝑁 + 1)))
2101, 15ssfid 9211 . . . . . . . . . . . 12 (𝜑𝐴 ∈ Fin)
211 fzfid 13878 . . . . . . . . . . . 12 (𝜑 → (1...(𝑁 + 1)) ∈ Fin)
212 hashen 14247 . . . . . . . . . . . 12 ((𝐴 ∈ Fin ∧ (1...(𝑁 + 1)) ∈ Fin) → ((♯‘𝐴) = (♯‘(1...(𝑁 + 1))) ↔ 𝐴 ≈ (1...(𝑁 + 1))))
213210, 211, 212syl2anc 584 . . . . . . . . . . 11 (𝜑 → ((♯‘𝐴) = (♯‘(1...(𝑁 + 1))) ↔ 𝐴 ≈ (1...(𝑁 + 1))))
214209, 213mpbird 256 . . . . . . . . . 10 (𝜑 → (♯‘𝐴) = (♯‘(1...(𝑁 + 1))))
215 hashfz1 14246 . . . . . . . . . . 11 ((𝑁 + 1) ∈ ℕ0 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
216203, 215syl 17 . . . . . . . . . 10 (𝜑 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
217214, 216eqtrd 2776 . . . . . . . . 9 (𝜑 → (♯‘𝐴) = (𝑁 + 1))
21827ex 413 . . . . . . . . . . . . . 14 (𝜑 → (𝑣𝐴 → ((𝑃 − 1) − 𝑣) ∈ (0...(𝑃 − 1))))
21920adantr 481 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣𝐴𝑘𝐴)) → (𝑃 − 1) ∈ ℂ)
220 fzssuz 13482 . . . . . . . . . . . . . . . . . . . 20 (0...(𝑃 − 1)) ⊆ (ℤ‘0)
221 uzssz 12784 . . . . . . . . . . . . . . . . . . . . 21 (ℤ‘0) ⊆ ℤ
222 zsscn 12507 . . . . . . . . . . . . . . . . . . . . 21 ℤ ⊆ ℂ
223221, 222sstri 3953 . . . . . . . . . . . . . . . . . . . 20 (ℤ‘0) ⊆ ℂ
224220, 223sstri 3953 . . . . . . . . . . . . . . . . . . 19 (0...(𝑃 − 1)) ⊆ ℂ
22515, 224sstrdi 3956 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ⊆ ℂ)
226225sselda 3944 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣𝐴) → 𝑣 ∈ ℂ)
227226adantrr 715 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣𝐴𝑘𝐴)) → 𝑣 ∈ ℂ)
228225sselda 3944 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → 𝑘 ∈ ℂ)
229228adantrl 714 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣𝐴𝑘𝐴)) → 𝑘 ∈ ℂ)
230219, 227, 229subcanad 11555 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑣𝐴𝑘𝐴)) → (((𝑃 − 1) − 𝑣) = ((𝑃 − 1) − 𝑘) ↔ 𝑣 = 𝑘))
231230ex 413 . . . . . . . . . . . . . 14 (𝜑 → ((𝑣𝐴𝑘𝐴) → (((𝑃 − 1) − 𝑣) = ((𝑃 − 1) − 𝑘) ↔ 𝑣 = 𝑘)))
232218, 231dom2lem 8932 . . . . . . . . . . . . 13 (𝜑 → (𝑣𝐴 ↦ ((𝑃 − 1) − 𝑣)):𝐴1-1→(0...(𝑃 − 1)))
233 f1eq1 6733 . . . . . . . . . . . . . 14 (𝐹 = (𝑣𝐴 ↦ ((𝑃 − 1) − 𝑣)) → (𝐹:𝐴1-1→(0...(𝑃 − 1)) ↔ (𝑣𝐴 ↦ ((𝑃 − 1) − 𝑣)):𝐴1-1→(0...(𝑃 − 1))))
23428, 233ax-mp 5 . . . . . . . . . . . . 13 (𝐹:𝐴1-1→(0...(𝑃 − 1)) ↔ (𝑣𝐴 ↦ ((𝑃 − 1) − 𝑣)):𝐴1-1→(0...(𝑃 − 1)))
235232, 234sylibr 233 . . . . . . . . . . . 12 (𝜑𝐹:𝐴1-1→(0...(𝑃 − 1)))
236 f1f1orn 6795 . . . . . . . . . . . 12 (𝐹:𝐴1-1→(0...(𝑃 − 1)) → 𝐹:𝐴1-1-onto→ran 𝐹)
237235, 236syl 17 . . . . . . . . . . 11 (𝜑𝐹:𝐴1-1-onto→ran 𝐹)
238210, 237hasheqf1od 14253 . . . . . . . . . 10 (𝜑 → (♯‘𝐴) = (♯‘ran 𝐹))
239238, 217eqtr3d 2778 . . . . . . . . 9 (𝜑 → (♯‘ran 𝐹) = (𝑁 + 1))
240217, 239oveq12d 7375 . . . . . . . 8 (𝜑 → ((♯‘𝐴) + (♯‘ran 𝐹)) = ((𝑁 + 1) + (𝑁 + 1)))
24159, 68, 2403eqtr4d 2786 . . . . . . 7 (𝜑 → (𝑃 + 1) = ((♯‘𝐴) + (♯‘ran 𝐹)))
242241adantr 481 . . . . . 6 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝑃 + 1) = ((♯‘𝐴) + (♯‘ran 𝐹)))
243210adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝐴 ∈ Fin)
2441, 30ssfid 9211 . . . . . . . 8 (𝜑 → ran 𝐹 ∈ Fin)
245244adantr 481 . . . . . . 7 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → ran 𝐹 ∈ Fin)
246 simpr 485 . . . . . . 7 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝐴 ∩ ran 𝐹) = ∅)
247 hashun 14282 . . . . . . 7 ((𝐴 ∈ Fin ∧ ran 𝐹 ∈ Fin ∧ (𝐴 ∩ ran 𝐹) = ∅) → (♯‘(𝐴 ∪ ran 𝐹)) = ((♯‘𝐴) + (♯‘ran 𝐹)))
248243, 245, 246, 247syl3anc 1371 . . . . . 6 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (♯‘(𝐴 ∪ ran 𝐹)) = ((♯‘𝐴) + (♯‘ran 𝐹)))
249242, 248eqtr4d 2779 . . . . 5 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝑃 + 1) = (♯‘(𝐴 ∪ ran 𝐹)))
25055, 249breqtrd 5131 . . . 4 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 < (♯‘(𝐴 ∪ ran 𝐹)))
251250ex 413 . . 3 (𝜑 → ((𝐴 ∩ ran 𝐹) = ∅ → 𝑃 < (♯‘(𝐴 ∪ ran 𝐹))))
252251necon3bd 2957 . 2 (𝜑 → (¬ 𝑃 < (♯‘(𝐴 ∪ ran 𝐹)) → (𝐴 ∩ ran 𝐹) ≠ ∅))
25353, 252mpd 15 1 (𝜑 → (𝐴 ∩ ran 𝐹) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 845   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wrex 3073  cun 3908  cin 3909  wss 3910  c0 4282   class class class wbr 5105  cmpt 5188  ran crn 5634  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  (class class class)co 7357  cen 8880  cdom 8881  Fincfn 8883  cc 11049  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056   < clt 11189  cle 11190  cmin 11385  cn 12153  2c2 12208  0cn0 12413  cz 12499  cuz 12763  ...cfz 13424   mod cmo 13774  cexp 13967  chash 14230  abscabs 15119  cdvds 16136  cprime 16547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-2o 8413  df-oadd 8416  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-dju 9837  df-card 9875  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-xnn0 12486  df-z 12500  df-uz 12764  df-rp 12916  df-fz 13425  df-fl 13697  df-mod 13775  df-seq 13907  df-exp 13968  df-hash 14231  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-dvds 16137  df-gcd 16375  df-prm 16548
This theorem is referenced by:  4sqlem12  16828
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