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Mirrors > Home > MPE Home > Th. List > Mathboxes > flt4lem2 | Structured version Visualization version GIF version |
Description: If 𝐴 is even, 𝐵 is odd. (Contributed by SN, 22-Aug-2024.) |
Ref | Expression |
---|---|
flt4lem2.a | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
flt4lem2.b | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
flt4lem2.c | ⊢ (𝜑 → 𝐶 ∈ ℕ) |
flt4lem2.1 | ⊢ (𝜑 → 2 ∥ 𝐴) |
flt4lem2.2 | ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) |
flt4lem2.3 | ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) |
Ref | Expression |
---|---|
flt4lem2 | ⊢ (𝜑 → ¬ 2 ∥ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flt4lem2.2 | . 2 ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) | |
2 | breq1 5033 | . . . . . . 7 ⊢ (𝑖 = 2 → (𝑖 ∥ 𝐴 ↔ 2 ∥ 𝐴)) | |
3 | breq1 5033 | . . . . . . 7 ⊢ (𝑖 = 2 → (𝑖 ∥ 𝐶 ↔ 2 ∥ 𝐶)) | |
4 | 2, 3 | anbi12d 634 | . . . . . 6 ⊢ (𝑖 = 2 → ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶) ↔ (2 ∥ 𝐴 ∧ 2 ∥ 𝐶))) |
5 | 2z 12095 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
6 | uzid 12339 | . . . . . . . 8 ⊢ (2 ∈ ℤ → 2 ∈ (ℤ≥‘2)) | |
7 | 5, 6 | ax-mp 5 | . . . . . . 7 ⊢ 2 ∈ (ℤ≥‘2) |
8 | 7 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∈ (ℤ≥‘2)) |
9 | flt4lem2.1 | . . . . . . . 8 ⊢ (𝜑 → 2 ∥ 𝐴) | |
10 | 9 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ 𝐴) |
11 | 5 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∈ ℤ) |
12 | flt4lem2.a | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
13 | flt4lem2.b | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
14 | gcdnncl 15950 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ) | |
15 | 12, 13, 14 | syl2anc 587 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ) |
16 | 15 | nnzd 12167 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ) |
17 | 16 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (𝐴 gcd 𝐵) ∈ ℤ) |
18 | flt4lem2.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
19 | 18 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐶 ∈ ℕ) |
20 | 19 | nnzd 12167 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐶 ∈ ℤ) |
21 | simpr 488 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ 𝐵) | |
22 | 12 | adantr 484 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐴 ∈ ℕ) |
23 | 22 | nnzd 12167 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐴 ∈ ℤ) |
24 | 13 | nnzd 12167 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
25 | 24 | adantr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐵 ∈ ℤ) |
26 | dvdsgcd 15988 | . . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ 2 ∥ 𝐵) → 2 ∥ (𝐴 gcd 𝐵))) | |
27 | 11, 23, 25, 26 | syl3anc 1372 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → ((2 ∥ 𝐴 ∧ 2 ∥ 𝐵) → 2 ∥ (𝐴 gcd 𝐵))) |
28 | 10, 21, 27 | mp2and 699 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ (𝐴 gcd 𝐵)) |
29 | 2nn 11789 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ | |
30 | 29 | a1i 11 | . . . . . . . . . 10 ⊢ (𝜑 → 2 ∈ ℕ) |
31 | flt4lem2.3 | . . . . . . . . . 10 ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) | |
32 | 12, 13, 18, 30, 31 | fltdvdsabdvdsc 40047 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶) |
33 | 32 | adantr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐶) |
34 | 11, 17, 20, 28, 33 | dvdstrd 15740 | . . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ 𝐶) |
35 | 10, 34 | jca 515 | . . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (2 ∥ 𝐴 ∧ 2 ∥ 𝐶)) |
36 | 4, 8, 35 | rspcedvdw 39771 | . . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → ∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶)) |
37 | ncoprmgcdne1b 16091 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶) ↔ (𝐴 gcd 𝐶) ≠ 1)) | |
38 | 22, 19, 37 | syl2anc 587 | . . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶) ↔ (𝐴 gcd 𝐶) ≠ 1)) |
39 | 36, 38 | mpbid 235 | . . . 4 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (𝐴 gcd 𝐶) ≠ 1) |
40 | 39 | ex 416 | . . 3 ⊢ (𝜑 → (2 ∥ 𝐵 → (𝐴 gcd 𝐶) ≠ 1)) |
41 | 40 | necon2bd 2950 | . 2 ⊢ (𝜑 → ((𝐴 gcd 𝐶) = 1 → ¬ 2 ∥ 𝐵)) |
42 | 1, 41 | mpd 15 | 1 ⊢ (𝜑 → ¬ 2 ∥ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 ∃wrex 3054 class class class wbr 5030 ‘cfv 6339 (class class class)co 7170 1c1 10616 + caddc 10618 ℕcn 11716 2c2 11771 ℤcz 12062 ℤ≥cuz 12324 ↑cexp 13521 ∥ cdvds 15699 gcd cgcd 15937 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 ax-pre-sup 10693 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-sup 8979 df-inf 8980 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-div 11376 df-nn 11717 df-2 11779 df-3 11780 df-n0 11977 df-z 12063 df-uz 12325 df-rp 12473 df-fl 13253 df-mod 13329 df-seq 13461 df-exp 13522 df-cj 14548 df-re 14549 df-im 14550 df-sqrt 14684 df-abs 14685 df-dvds 15700 df-gcd 15938 |
This theorem is referenced by: flt4lem3 40057 flt4lem7 40068 nna4b4nsq 40069 |
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