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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 5142 | . . . . . . 7 ⊢ (𝑖 = 2 → (𝑖 ∥ 𝐴 ↔ 2 ∥ 𝐴)) | |
3 | breq1 5142 | . . . . . . 7 ⊢ (𝑖 = 2 → (𝑖 ∥ 𝐶 ↔ 2 ∥ 𝐶)) | |
4 | 2, 3 | anbi12d 630 | . . . . . 6 ⊢ (𝑖 = 2 → ((𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶) ↔ (2 ∥ 𝐴 ∧ 2 ∥ 𝐶))) |
5 | 2z 12593 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
6 | uzid 12836 | . . . . . . . 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 480 | . . . . . . 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 16451 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ) | |
15 | 12, 13, 14 | syl2anc 583 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ) |
16 | 15 | nnzd 12584 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ) |
17 | 16 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (𝐴 gcd 𝐵) ∈ ℤ) |
18 | flt4lem2.c | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
19 | 18 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐶 ∈ ℕ) |
20 | 19 | nnzd 12584 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐶 ∈ ℤ) |
21 | simpr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ 𝐵) | |
22 | 12 | adantr 480 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐴 ∈ ℕ) |
23 | 22 | nnzd 12584 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐴 ∈ ℤ) |
24 | 13 | nnzd 12584 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
25 | 24 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐵 ∈ ℤ) |
26 | dvdsgcd 16489 | . . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ 2 ∥ 𝐵) → 2 ∥ (𝐴 gcd 𝐵))) | |
27 | 11, 23, 25, 26 | syl3anc 1368 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → ((2 ∥ 𝐴 ∧ 2 ∥ 𝐵) → 2 ∥ (𝐴 gcd 𝐵))) |
28 | 10, 21, 27 | mp2and 696 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ (𝐴 gcd 𝐵)) |
29 | 2nn 12284 | . . . . . . . . . . 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 41930 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶) |
33 | 32 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (𝐴 gcd 𝐵) ∥ 𝐶) |
34 | 11, 17, 20, 28, 33 | dvdstrd 16241 | . . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ 𝐶) |
35 | 10, 34 | jca 511 | . . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (2 ∥ 𝐴 ∧ 2 ∥ 𝐶)) |
36 | 4, 8, 35 | rspcedvdw 3607 | . . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → ∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶)) |
37 | ncoprmgcdne1b 16590 | . . . . . 6 ⊢ ((𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) → (∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶) ↔ (𝐴 gcd 𝐶) ≠ 1)) | |
38 | 22, 19, 37 | syl2anc 583 | . . . . 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 412 | . . 3 ⊢ (𝜑 → (2 ∥ 𝐵 → (𝐴 gcd 𝐶) ≠ 1)) |
41 | 40 | necon2bd 2948 | . 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 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∃wrex 3062 class class class wbr 5139 ‘cfv 6534 (class class class)co 7402 1c1 11108 + caddc 11110 ℕcn 12211 2c2 12266 ℤcz 12557 ℤ≥cuz 12821 ↑cexp 14028 ∥ cdvds 16200 gcd cgcd 16438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 ax-pre-sup 11185 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-iun 4990 df-br 5140 df-opab 5202 df-mpt 5223 df-tr 5257 df-id 5565 df-eprel 5571 df-po 5579 df-so 5580 df-fr 5622 df-we 5624 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6291 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-riota 7358 df-ov 7405 df-oprab 7406 df-mpo 7407 df-om 7850 df-2nd 7970 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-sup 9434 df-inf 9435 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11445 df-neg 11446 df-div 11871 df-nn 12212 df-2 12274 df-3 12275 df-n0 12472 df-z 12558 df-uz 12822 df-rp 12976 df-fl 13758 df-mod 13836 df-seq 13968 df-exp 14029 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-dvds 16201 df-gcd 16439 |
This theorem is referenced by: flt4lem3 41940 flt4lem7 41951 nna4b4nsq 41952 |
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