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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 5145 | . . . . . . 7 ⊢ (𝑖 = 2 → (𝑖 ∥ 𝐴 ↔ 2 ∥ 𝐴)) | |
3 | breq1 5145 | . . . . . . 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 12616 | . . . . . . . 8 ⊢ 2 ∈ ℤ | |
6 | uzid 12859 | . . . . . . . 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 16473 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ) | |
15 | 12, 13, 14 | syl2anc 583 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ) |
16 | 15 | nnzd 12607 | . . . . . . . . 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 12607 | . . . . . . . 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 12607 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐴 ∈ ℤ) |
24 | 13 | nnzd 12607 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
25 | 24 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 𝐵 ∈ ℤ) |
26 | dvdsgcd 16511 | . . . . . . . . . 10 ⊢ ((2 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((2 ∥ 𝐴 ∧ 2 ∥ 𝐵) → 2 ∥ (𝐴 gcd 𝐵))) | |
27 | 11, 23, 25, 26 | syl3anc 1369 | . . . . . . . . 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 12307 | . . . . . . . . . . 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 41984 | . . . . . . . . 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 16263 | . . . . . . 7 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → 2 ∥ 𝐶) |
35 | 10, 34 | jca 511 | . . . . . 6 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → (2 ∥ 𝐴 ∧ 2 ∥ 𝐶)) |
36 | 4, 8, 35 | rspcedvdw 3610 | . . . . 5 ⊢ ((𝜑 ∧ 2 ∥ 𝐵) → ∃𝑖 ∈ (ℤ≥‘2)(𝑖 ∥ 𝐴 ∧ 𝑖 ∥ 𝐶)) |
37 | ncoprmgcdne1b 16612 | . . . . . 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 2951 | . 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 1534 ∈ wcel 2099 ≠ wne 2935 ∃wrex 3065 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 1c1 11131 + caddc 11133 ℕcn 12234 2c2 12289 ℤcz 12580 ℤ≥cuz 12844 ↑cexp 14050 ∥ cdvds 16222 gcd cgcd 16460 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-sup 9457 df-inf 9458 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-rp 12999 df-fl 13781 df-mod 13859 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-dvds 16223 df-gcd 16461 |
This theorem is referenced by: flt4lem3 41994 flt4lem7 42005 nna4b4nsq 42006 |
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