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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 5112 | . . . . . . 7 ⊢ (𝑖 = 2 → (𝑖 ∥ 𝐴 ↔ 2 ∥ 𝐴)) | |
3 | breq1 5112 | . . . . . . 7 ⊢ (𝑖 = 2 → (𝑖 ∥ 𝐶 ↔ 2 ∥ 𝐶)) | |
4 | 2, 3 | anbi12d 632 | . . . . . 6 ⊢ (𝑖 = 2 → ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶) ↔ (2 ∥ 𝐴 ∧ 2 ∥ 𝐶))) |
5 | 2z 12543 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
6 | uzid 12786 | . . . . . . . 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 482 | . . . . . . 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 16395 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ) | |
15 | 12, 13, 14 | syl2anc 585 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ) |
16 | 15 | nnzd 12534 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ) |
17 | 16 | adantr 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (𝐴 gcd 𝐵) ∈ ℤ) |
18 | flt4lem2.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
19 | 18 | adantr 482 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐶 ∈ ℕ) |
20 | 19 | nnzd 12534 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐶 ∈ ℤ) |
21 | simpr 486 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ 𝐵) | |
22 | 12 | adantr 482 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐴 ∈ ℕ) |
23 | 22 | nnzd 12534 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐴 ∈ ℤ) |
24 | 13 | nnzd 12534 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
25 | 24 | adantr 482 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐵 ∈ ℤ) |
26 | dvdsgcd 16433 | . . . . . . . . . 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 698 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ (𝐴 gcd 𝐵)) |
29 | 2nn 12234 | . . . . . . . . . . 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 41023 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶) |
33 | 32 | adantr 482 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐶) |
34 | 11, 17, 20, 28, 33 | dvdstrd 16185 | . . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ 𝐶) |
35 | 10, 34 | jca 513 | . . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (2 ∥ 𝐴 ∧ 2 ∥ 𝐶)) |
36 | 4, 8, 35 | rspcedvdw 40681 | . . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → ∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶)) |
37 | ncoprmgcdne1b 16534 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶) ↔ (𝐴 gcd 𝐶) ≠ 1)) | |
38 | 22, 19, 37 | syl2anc 585 | . . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶) ↔ (𝐴 gcd 𝐶) ≠ 1)) |
39 | 36, 38 | mpbid 231 | . . . 4 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (𝐴 gcd 𝐶) ≠ 1) |
40 | 39 | ex 414 | . . 3 ⊢ (𝜑 → (2 ∥ 𝐵 → (𝐴 gcd 𝐶) ≠ 1)) |
41 | 40 | necon2bd 2956 | . 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∃wrex 3070 class class class wbr 5109 ‘cfv 6500 (class class class)co 7361 1c1 11060 + caddc 11062 ℕcn 12161 2c2 12216 ℤcz 12507 ℤ≥cuz 12771 ↑cexp 13976 ∥ cdvds 16144 gcd cgcd 16382 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-sup 9386 df-inf 9387 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-n0 12422 df-z 12508 df-uz 12772 df-rp 12924 df-fl 13706 df-mod 13784 df-seq 13916 df-exp 13977 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-dvds 16145 df-gcd 16383 |
This theorem is referenced by: flt4lem3 41033 flt4lem7 41044 nna4b4nsq 41045 |
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