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Theorem 4sqlem11 16993
Description: Lemma for 4sq 17002. 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 14014 . . . . . 6 (𝜑 → (0...(𝑃 − 1)) ∈ Fin)
2 4sqlem11.5 . . . . . . . 8 𝐴 = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
3 elfzelz 13564 . . . . . . . . . . . . 13 (𝑚 ∈ (0...𝑁) → 𝑚 ∈ ℤ)
4 zsqcl 14169 . . . . . . . . . . . . 13 (𝑚 ∈ ℤ → (𝑚↑2) ∈ ℤ)
53, 4syl 17 . . . . . . . . . . . 12 (𝑚 ∈ (0...𝑁) → (𝑚↑2) ∈ ℤ)
6 4sq.4 . . . . . . . . . . . . 13 (𝜑𝑃 ∈ ℙ)
7 prmnn 16711 . . . . . . . . . . . . 13 (𝑃 ∈ ℙ → 𝑃 ∈ ℕ)
86, 7syl 17 . . . . . . . . . . . 12 (𝜑𝑃 ∈ ℕ)
9 zmodfz 13933 . . . . . . . . . . . 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 2836 . . . . . . . . . . 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 3155 . . . . . . . . 9 (𝜑 → (∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃) → 𝑢 ∈ (0...(𝑃 − 1))))
1413abssdv 4068 . . . . . . . 8 (𝜑 → {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)} ⊆ (0...(𝑃 − 1)))
152, 14eqsstrid 4022 . . . . . . 7 (𝜑𝐴 ⊆ (0...(𝑃 − 1)))
16 prmz 16712 . . . . . . . . . . . . . . . 16 (𝑃 ∈ ℙ → 𝑃 ∈ ℤ)
176, 16syl 17 . . . . . . . . . . . . . . 15 (𝜑𝑃 ∈ ℤ)
18 peano2zm 12660 . . . . . . . . . . . . . . 15 (𝑃 ∈ ℤ → (𝑃 − 1) ∈ ℤ)
1917, 18syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑃 − 1) ∈ ℤ)
2019zcnd 12723 . . . . . . . . . . . . 13 (𝜑 → (𝑃 − 1) ∈ ℂ)
2120addlidd 11462 . . . . . . . . . . . 12 (𝜑 → (0 + (𝑃 − 1)) = (𝑃 − 1))
2221oveq1d 7446 . . . . . . . . . . 11 (𝜑 → ((0 + (𝑃 − 1)) − 𝑣) = ((𝑃 − 1) − 𝑣))
2322adantr 480 . . . . . . . . . 10 ((𝜑𝑣𝐴) → ((0 + (𝑃 − 1)) − 𝑣) = ((𝑃 − 1) − 𝑣))
2415sselda 3983 . . . . . . . . . . 11 ((𝜑𝑣𝐴) → 𝑣 ∈ (0...(𝑃 − 1)))
25 fzrev3i 13631 . . . . . . . . . . 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 2842 . . . . . . . . 9 ((𝜑𝑣𝐴) → ((𝑃 − 1) − 𝑣) ∈ (0...(𝑃 − 1)))
28 4sqlem11.6 . . . . . . . . 9 𝐹 = (𝑣𝐴 ↦ ((𝑃 − 1) − 𝑣))
2927, 28fmptd 7134 . . . . . . . 8 (𝜑𝐹:𝐴⟶(0...(𝑃 − 1)))
3029frnd 6744 . . . . . . 7 (𝜑 → ran 𝐹 ⊆ (0...(𝑃 − 1)))
3115, 30unssd 4192 . . . . . 6 (𝜑 → (𝐴 ∪ ran 𝐹) ⊆ (0...(𝑃 − 1)))
321, 31ssfid 9301 . . . . 5 (𝜑 → (𝐴 ∪ ran 𝐹) ∈ Fin)
33 hashcl 14395 . . . . 5 ((𝐴 ∪ ran 𝐹) ∈ Fin → (♯‘(𝐴 ∪ ran 𝐹)) ∈ ℕ0)
3432, 33syl 17 . . . 4 (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ∈ ℕ0)
3534nn0red 12588 . . 3 (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ∈ ℝ)
3617zred 12722 . . 3 (𝜑𝑃 ∈ ℝ)
37 ssdomg 9040 . . . . . 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 14418 . . . . . 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 257 . . . 4 (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ≤ (♯‘(0...(𝑃 − 1))))
42 fz01en 13592 . . . . . . 7 (𝑃 ∈ ℤ → (0...(𝑃 − 1)) ≈ (1...𝑃))
4317, 42syl 17 . . . . . 6 (𝜑 → (0...(𝑃 − 1)) ≈ (1...𝑃))
44 fzfid 14014 . . . . . . 7 (𝜑 → (1...𝑃) ∈ Fin)
45 hashen 14386 . . . . . . 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 257 . . . . 5 (𝜑 → (♯‘(0...(𝑃 − 1))) = (♯‘(1...𝑃)))
488nnnn0d 12587 . . . . . 6 (𝜑𝑃 ∈ ℕ0)
49 hashfz1 14385 . . . . . 6 (𝑃 ∈ ℕ0 → (♯‘(1...𝑃)) = 𝑃)
5048, 49syl 17 . . . . 5 (𝜑 → (♯‘(1...𝑃)) = 𝑃)
5147, 50eqtrd 2777 . . . 4 (𝜑 → (♯‘(0...(𝑃 − 1))) = 𝑃)
5241, 51breqtrd 5169 . . 3 (𝜑 → (♯‘(𝐴 ∪ ran 𝐹)) ≤ 𝑃)
5335, 36, 52lensymd 11412 . 2 (𝜑 → ¬ 𝑃 < (♯‘(𝐴 ∪ ran 𝐹)))
5436adantr 480 . . . . . 6 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 ∈ ℝ)
5554ltp1d 12198 . . . . 5 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 < (𝑃 + 1))
56 4sq.2 . . . . . . . . . 10 (𝜑𝑁 ∈ ℕ)
5756nncnd 12282 . . . . . . . . 9 (𝜑𝑁 ∈ ℂ)
58 1cnd 11256 . . . . . . . . 9 (𝜑 → 1 ∈ ℂ)
5957, 57, 58, 58add4d 11490 . . . . . . . 8 (𝜑 → ((𝑁 + 𝑁) + (1 + 1)) = ((𝑁 + 1) + (𝑁 + 1)))
60 4sq.3 . . . . . . . . . 10 (𝜑𝑃 = ((2 · 𝑁) + 1))
6160oveq1d 7446 . . . . . . . . 9 (𝜑 → (𝑃 + 1) = (((2 · 𝑁) + 1) + 1))
62 2cn 12341 . . . . . . . . . . 11 2 ∈ ℂ
63 mulcl 11239 . . . . . . . . . . 11 ((2 ∈ ℂ ∧ 𝑁 ∈ ℂ) → (2 · 𝑁) ∈ ℂ)
6462, 57, 63sylancr 587 . . . . . . . . . 10 (𝜑 → (2 · 𝑁) ∈ ℂ)
6564, 58, 58addassd 11283 . . . . . . . . 9 (𝜑 → (((2 · 𝑁) + 1) + 1) = ((2 · 𝑁) + (1 + 1)))
66572timesd 12509 . . . . . . . . . 10 (𝜑 → (2 · 𝑁) = (𝑁 + 𝑁))
6766oveq1d 7446 . . . . . . . . 9 (𝜑 → ((2 · 𝑁) + (1 + 1)) = ((𝑁 + 𝑁) + (1 + 1)))
6861, 65, 673eqtrd 2781 . . . . . . . 8 (𝜑 → (𝑃 + 1) = ((𝑁 + 𝑁) + (1 + 1)))
6910ex 412 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑚 ∈ (0...𝑁) → ((𝑚↑2) mod 𝑃) ∈ (0...(𝑃 − 1))))
708adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℕ)
713ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . 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 13564 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑢 ∈ (0...𝑁) → 𝑢 ∈ ℤ)
7473ad2antll 729 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℤ)
75 zsqcl 14169 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑢 ∈ ℤ → (𝑢↑2) ∈ ℤ)
7674, 75syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑢↑2) ∈ ℤ)
77 moddvds 16301 . . . . . . . . . . . . . . . . . . . . . 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 1373 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑃 ∥ ((𝑚↑2) − (𝑢↑2))))
7971zcnd 12723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℂ)
8074zcnd 12723 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℂ)
81 subsq 14249 . . . . . . . . . . . . . . . . . . . . . . 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 5155 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ ((𝑚↑2) − (𝑢↑2)) ↔ 𝑃 ∥ ((𝑚 + 𝑢) · (𝑚𝑢))))
846adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℙ)
8571, 74zaddcld 12726 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ∈ ℤ)
8671, 74zsubcld 12727 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚𝑢) ∈ ℤ)
87 euclemma 16750 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑃 ∈ ℙ ∧ (𝑚 + 𝑢) ∈ ℤ ∧ (𝑚𝑢) ∈ ℤ) → (𝑃 ∥ ((𝑚 + 𝑢) · (𝑚𝑢)) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚𝑢))))
8884, 85, 86, 87syl3anc 1373 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ ((𝑚 + 𝑢) · (𝑚𝑢)) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚𝑢))))
8978, 83, 883bitrd 305 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ (𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚𝑢))))
9085zred 12722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ∈ ℝ)
91 2re 12340 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2 ∈ ℝ
9256nnred 12281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑𝑁 ∈ ℝ)
93 remulcl 11240 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((2 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (2 · 𝑁) ∈ ℝ)
9491, 92, 93sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (2 · 𝑁) ∈ ℝ)
9594adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑃 ∈ ℝ)
9871zred 12722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚 ∈ ℝ)
9974zred 12722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢 ∈ ℝ)
10092adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑁 ∈ ℝ)
101 elfzle2 13568 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑚 ∈ (0...𝑁) → 𝑚𝑁)
102101ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑚𝑁)
103 elfzle2 13568 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑢 ∈ (0...𝑁) → 𝑢𝑁)
104103ad2antll 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑢𝑁)
10598, 99, 100, 100, 102, 104le2addd 11882 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ≤ (𝑁 + 𝑁))
10657adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 𝑁 ∈ ℂ)
1071062timesd 12509 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) = (𝑁 + 𝑁))
108105, 107breqtrrd 5171 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) ≤ (2 · 𝑁))
10994ltp1d 12198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝜑 → (2 · 𝑁) < ((2 · 𝑁) + 1))
110109, 60breqtrrd 5171 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝜑 → (2 · 𝑁) < 𝑃)
111110adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (2 · 𝑁) < 𝑃)
11290, 95, 97, 108, 111lelttrd 11419 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 + 𝑢) < 𝑃)
11390, 97ltnled 11408 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚 + 𝑢) < 𝑃 ↔ ¬ 𝑃 ≤ (𝑚 + 𝑢)))
114112, 113mpbid 232 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ¬ 𝑃 ≤ (𝑚 + 𝑢))
115114adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → ¬ 𝑃 ≤ (𝑚 + 𝑢))
11617ad2antrr 726 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 𝑃 ∈ ℤ)
11785adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑚 + 𝑢) ∈ ℤ)
118 1red 11262 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 1 ∈ ℝ)
119 nn0abscl 15351 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑚𝑢) ∈ ℤ → (abs‘(𝑚𝑢)) ∈ ℕ0)
12086, 119syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚𝑢)) ∈ ℕ0)
121120nn0red 12588 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚𝑢)) ∈ ℝ)
122121adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ∈ ℝ)
123117zred 12722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑚 + 𝑢) ∈ ℝ)
124120adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ∈ ℕ0)
125124nn0zd 12639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ∈ ℤ)
12686zcnd 12723 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚𝑢) ∈ ℂ)
127126adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑚𝑢) ∈ ℂ)
12879, 80subeq0ad 11630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚𝑢) = 0 ↔ 𝑚 = 𝑢))
129128necon3bid 2985 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑚𝑢) ≠ 0 ↔ 𝑚𝑢))
130129biimpar 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (𝑚𝑢) ≠ 0)
131127, 130absrpcld 15487 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ∈ ℝ+)
132131rpgt0d 13080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 0 < (abs‘(𝑚𝑢)))
133 elnnz 12623 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12314 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 1 ≤ (abs‘(𝑚𝑢)))
136 0cnd 11254 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ∈ ℂ)
13779, 80, 136abs3difd 15499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚𝑢)) ≤ ((abs‘(𝑚 − 0)) + (abs‘(0 − 𝑢))))
13879subid1d 11609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚 − 0) = 𝑚)
139138fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 0)) = (abs‘𝑚))
140 elfzle1 13567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑚 ∈ (0...𝑁) → 0 ≤ 𝑚)
141140ad2antrl 728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ≤ 𝑚)
14298, 141absidd 15461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘𝑚) = 𝑚)
143139, 142eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚 − 0)) = 𝑚)
144 0cn 11253 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 0 ∈ ℂ
145 abssub 15365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑢 − 0) = 𝑢)
148147fveq2d 6910 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑢 − 0)) = (abs‘𝑢))
149 elfzle1 13567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑢 ∈ (0...𝑁) → 0 ≤ 𝑢)
150149ad2antll 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → 0 ≤ 𝑢)
15199, 150absidd 15461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘𝑢) = 𝑢)
152146, 148, 1513eqtrd 2781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(0 − 𝑢)) = 𝑢)
153143, 152oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((abs‘(𝑚 − 0)) + (abs‘(0 − 𝑢))) = (𝑚 + 𝑢))
154137, 153breqtrd 5169 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (abs‘(𝑚𝑢)) ≤ (𝑚 + 𝑢))
155154adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → (abs‘(𝑚𝑢)) ≤ (𝑚 + 𝑢))
156118, 122, 123, 135, 155letrd 11418 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 1 ≤ (𝑚 + 𝑢))
157 elnnz1 12643 . . . . . . . . . . . . . . . . . . . . . . . . . 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 16347 . . . . . . . . . . . . . . . . . . . . . . . . 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 198 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → ¬ 𝑃 ∥ (𝑚 + 𝑢))
162161ex 412 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚𝑢 → ¬ 𝑃 ∥ (𝑚 + 𝑢)))
163162necon4ad 2959 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚 + 𝑢) → 𝑚 = 𝑢))
164 dvdsabsb 16313 . . . . . . . . . . . . . . . . . . . . . . 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 11355 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑃 ∈ ℝ ∧ (abs‘(𝑚𝑢)) ∈ ℝ ∧ (𝑚 + 𝑢) ∈ ℝ) → ((𝑃 ≤ (abs‘(𝑚𝑢)) ∧ (abs‘(𝑚𝑢)) ≤ (𝑚 + 𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
16797, 121, 90, 166syl3anc 1373 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑃 ≤ (abs‘(𝑚𝑢)) ∧ (abs‘(𝑚𝑢)) ≤ (𝑚 + 𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
168154, 167mpan2d 694 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ≤ (abs‘(𝑚𝑢)) → 𝑃 ≤ (𝑚 + 𝑢)))
169114, 168mtod 198 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ¬ 𝑃 ≤ (abs‘(𝑚𝑢)))
170169adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → ¬ 𝑃 ≤ (abs‘(𝑚𝑢)))
17196adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → 𝑃 ∈ ℤ)
172 dvdsle 16347 . . . . . . . . . . . . . . . . . . . . . . . . . 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 198 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) ∧ 𝑚𝑢) → ¬ 𝑃 ∥ (abs‘(𝑚𝑢)))
175174ex 412 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑚𝑢 → ¬ 𝑃 ∥ (abs‘(𝑚𝑢))))
176175necon4ad 2959 . . . . . . . . . . . . . . . . . . . . . 22 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (abs‘(𝑚𝑢)) → 𝑚 = 𝑢))
177165, 176sylbid 240 . . . . . . . . . . . . . . . . . . . . 21 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (𝑃 ∥ (𝑚𝑢) → 𝑚 = 𝑢))
178163, 177jaod 860 . . . . . . . . . . . . . . . . . . . 20 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → ((𝑃 ∥ (𝑚 + 𝑢) ∨ 𝑃 ∥ (𝑚𝑢)) → 𝑚 = 𝑢))
17989, 178sylbid 240 . . . . . . . . . . . . . . . . . . 19 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) → 𝑚 = 𝑢))
180 oveq1 7438 . . . . . . . . . . . . . . . . . . . 20 (𝑚 = 𝑢 → (𝑚↑2) = (𝑢↑2))
181180oveq1d 7446 . . . . . . . . . . . . . . . . . . 19 (𝑚 = 𝑢 → ((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃))
182179, 181impbid1 225 . . . . . . . . . . . . . . . . . 18 ((𝜑 ∧ (𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁))) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑚 = 𝑢))
183182ex 412 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑚 ∈ (0...𝑁) ∧ 𝑢 ∈ (0...𝑁)) → (((𝑚↑2) mod 𝑃) = ((𝑢↑2) mod 𝑃) ↔ 𝑚 = 𝑢)))
18469, 183dom2lem 9032 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1→(0...(𝑃 − 1)))
185 f1f1orn 6859 . . . . . . . . . . . . . . . 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 2737 . . . . . . . . . . . . . . . . . 18 (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)) = (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃))
188187rnmpt 5968 . . . . . . . . . . . . . . . . 17 ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)) = {𝑢 ∣ ∃𝑚 ∈ (0...𝑁)𝑢 = ((𝑚↑2) mod 𝑃)}
1892, 188eqtr4i 2768 . . . . . . . . . . . . . . . 16 𝐴 = ran (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃))
190 f1oeq3 6838 . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . 14 (𝜑 → (𝑚 ∈ (0...𝑁) ↦ ((𝑚↑2) mod 𝑃)):(0...𝑁)–1-1-onto𝐴)
193 ovex 7464 . . . . . . . . . . . . . . 15 (0...𝑁) ∈ V
194193f1oen 9013 . . . . . . . . . . . . . 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 9045 . . . . . . . . . . . 12 (𝜑𝐴 ≈ (0...𝑁))
197 ax-1cn 11213 . . . . . . . . . . . . . . 15 1 ∈ ℂ
198 pncan 11514 . . . . . . . . . . . . . . 15 ((𝑁 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑁 + 1) − 1) = 𝑁)
19957, 197, 198sylancl 586 . . . . . . . . . . . . . 14 (𝜑 → ((𝑁 + 1) − 1) = 𝑁)
200199oveq2d 7447 . . . . . . . . . . . . 13 (𝜑 → (0...((𝑁 + 1) − 1)) = (0...𝑁))
20156nnnn0d 12587 . . . . . . . . . . . . . . . 16 (𝜑𝑁 ∈ ℕ0)
202 peano2nn0 12566 . . . . . . . . . . . . . . . 16 (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0)
203201, 202syl 17 . . . . . . . . . . . . . . 15 (𝜑 → (𝑁 + 1) ∈ ℕ0)
204203nn0zd 12639 . . . . . . . . . . . . . 14 (𝜑 → (𝑁 + 1) ∈ ℤ)
205 fz01en 13592 . . . . . . . . . . . . . 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 5167 . . . . . . . . . . . 12 (𝜑 → (0...𝑁) ≈ (1...(𝑁 + 1)))
208 entr 9046 . . . . . . . . . . . 12 ((𝐴 ≈ (0...𝑁) ∧ (0...𝑁) ≈ (1...(𝑁 + 1))) → 𝐴 ≈ (1...(𝑁 + 1)))
209196, 207, 208syl2anc 584 . . . . . . . . . . 11 (𝜑𝐴 ≈ (1...(𝑁 + 1)))
2101, 15ssfid 9301 . . . . . . . . . . . 12 (𝜑𝐴 ∈ Fin)
211 fzfid 14014 . . . . . . . . . . . 12 (𝜑 → (1...(𝑁 + 1)) ∈ Fin)
212 hashen 14386 . . . . . . . . . . . 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 257 . . . . . . . . . 10 (𝜑 → (♯‘𝐴) = (♯‘(1...(𝑁 + 1))))
215 hashfz1 14385 . . . . . . . . . . 11 ((𝑁 + 1) ∈ ℕ0 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
216203, 215syl 17 . . . . . . . . . 10 (𝜑 → (♯‘(1...(𝑁 + 1))) = (𝑁 + 1))
217214, 216eqtrd 2777 . . . . . . . . 9 (𝜑 → (♯‘𝐴) = (𝑁 + 1))
21827ex 412 . . . . . . . . . . . . . 14 (𝜑 → (𝑣𝐴 → ((𝑃 − 1) − 𝑣) ∈ (0...(𝑃 − 1))))
21920adantr 480 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣𝐴𝑘𝐴)) → (𝑃 − 1) ∈ ℂ)
220 fzssuz 13605 . . . . . . . . . . . . . . . . . . . 20 (0...(𝑃 − 1)) ⊆ (ℤ‘0)
221 uzssz 12899 . . . . . . . . . . . . . . . . . . . . 21 (ℤ‘0) ⊆ ℤ
222 zsscn 12621 . . . . . . . . . . . . . . . . . . . . 21 ℤ ⊆ ℂ
223221, 222sstri 3993 . . . . . . . . . . . . . . . . . . . 20 (ℤ‘0) ⊆ ℂ
224220, 223sstri 3993 . . . . . . . . . . . . . . . . . . 19 (0...(𝑃 − 1)) ⊆ ℂ
22515, 224sstrdi 3996 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴 ⊆ ℂ)
226225sselda 3983 . . . . . . . . . . . . . . . . 17 ((𝜑𝑣𝐴) → 𝑣 ∈ ℂ)
227226adantrr 717 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣𝐴𝑘𝐴)) → 𝑣 ∈ ℂ)
228225sselda 3983 . . . . . . . . . . . . . . . . 17 ((𝜑𝑘𝐴) → 𝑘 ∈ ℂ)
229228adantrl 716 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝑣𝐴𝑘𝐴)) → 𝑘 ∈ ℂ)
230219, 227, 229subcanad 11663 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑣𝐴𝑘𝐴)) → (((𝑃 − 1) − 𝑣) = ((𝑃 − 1) − 𝑘) ↔ 𝑣 = 𝑘))
231230ex 412 . . . . . . . . . . . . . 14 (𝜑 → ((𝑣𝐴𝑘𝐴) → (((𝑃 − 1) − 𝑣) = ((𝑃 − 1) − 𝑘) ↔ 𝑣 = 𝑘)))
232218, 231dom2lem 9032 . . . . . . . . . . . . 13 (𝜑 → (𝑣𝐴 ↦ ((𝑃 − 1) − 𝑣)):𝐴1-1→(0...(𝑃 − 1)))
233 f1eq1 6799 . . . . . . . . . . . . . 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 234 . . . . . . . . . . . 12 (𝜑𝐹:𝐴1-1→(0...(𝑃 − 1)))
236 f1f1orn 6859 . . . . . . . . . . . 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 14392 . . . . . . . . . 10 (𝜑 → (♯‘𝐴) = (♯‘ran 𝐹))
239238, 217eqtr3d 2779 . . . . . . . . 9 (𝜑 → (♯‘ran 𝐹) = (𝑁 + 1))
240217, 239oveq12d 7449 . . . . . . . 8 (𝜑 → ((♯‘𝐴) + (♯‘ran 𝐹)) = ((𝑁 + 1) + (𝑁 + 1)))
24159, 68, 2403eqtr4d 2787 . . . . . . 7 (𝜑 → (𝑃 + 1) = ((♯‘𝐴) + (♯‘ran 𝐹)))
242241adantr 480 . . . . . 6 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝑃 + 1) = ((♯‘𝐴) + (♯‘ran 𝐹)))
243210adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝐴 ∈ Fin)
2441, 30ssfid 9301 . . . . . . . 8 (𝜑 → ran 𝐹 ∈ Fin)
245244adantr 480 . . . . . . 7 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → ran 𝐹 ∈ Fin)
246 simpr 484 . . . . . . 7 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝐴 ∩ ran 𝐹) = ∅)
247 hashun 14421 . . . . . . 7 ((𝐴 ∈ Fin ∧ ran 𝐹 ∈ Fin ∧ (𝐴 ∩ ran 𝐹) = ∅) → (♯‘(𝐴 ∪ ran 𝐹)) = ((♯‘𝐴) + (♯‘ran 𝐹)))
248243, 245, 246, 247syl3anc 1373 . . . . . 6 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (♯‘(𝐴 ∪ ran 𝐹)) = ((♯‘𝐴) + (♯‘ran 𝐹)))
249242, 248eqtr4d 2780 . . . . 5 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → (𝑃 + 1) = (♯‘(𝐴 ∪ ran 𝐹)))
25055, 249breqtrd 5169 . . . 4 ((𝜑 ∧ (𝐴 ∩ ran 𝐹) = ∅) → 𝑃 < (♯‘(𝐴 ∪ ran 𝐹)))
251250ex 412 . . 3 (𝜑 → ((𝐴 ∩ ran 𝐹) = ∅ → 𝑃 < (♯‘(𝐴 ∪ ran 𝐹))))
252251necon3bd 2954 . 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 206  wa 395  wo 848   = wceq 1540  wcel 2108  {cab 2714  wne 2940  wrex 3070  cun 3949  cin 3950  wss 3951  c0 4333   class class class wbr 5143  cmpt 5225  ran crn 5686  1-1wf1 6558  1-1-ontowf1o 6560  cfv 6561  (class class class)co 7431  cen 8982  cdom 8983  Fincfn 8985  cc 11153  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295  cle 11296  cmin 11492  cn 12266  2c2 12321  0cn0 12526  cz 12613  cuz 12878  ...cfz 13547   mod cmo 13909  cexp 14102  chash 14369  abscabs 15273  cdvds 16290  cprime 16708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-inf 9483  df-dju 9941  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fl 13832  df-mod 13910  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-dvds 16291  df-gcd 16532  df-prm 16709
This theorem is referenced by:  4sqlem12  16994
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