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Mirrors > Home > MPE Home > Th. List > Mathboxes > flt4lem1 | Structured version Visualization version GIF version |
Description: Satisfy the antecedent used in several pythagtrip 16867 lemmas, with 𝐴, 𝐶 coprime rather than 𝐴, 𝐵. (Contributed by SN, 21-Aug-2024.) |
Ref | Expression |
---|---|
flt4lem1.a | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
flt4lem1.b | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
flt4lem1.c | ⊢ (𝜑 → 𝐶 ∈ ℕ) |
flt4lem1.1 | ⊢ (𝜑 → ¬ 2 ∥ 𝐴) |
flt4lem1.2 | ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) |
flt4lem1.3 | ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) |
Ref | Expression |
---|---|
flt4lem1 | ⊢ (𝜑 → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flt4lem1.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | flt4lem1.b | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
3 | flt4lem1.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
4 | 1, 2, 3 | 3jca 1127 | . 2 ⊢ (𝜑 → (𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ)) |
5 | flt4lem1.3 | . 2 ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) | |
6 | flt4lem1.2 | . . . 4 ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) | |
7 | 1, 2, 3, 6, 5 | fltabcoprm 42628 | . . 3 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) |
8 | flt4lem1.1 | . . 3 ⊢ (𝜑 → ¬ 2 ∥ 𝐴) | |
9 | 7, 8 | jca 511 | . 2 ⊢ (𝜑 → ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴)) |
10 | 4, 5, 9 | 3jca 1127 | 1 ⊢ (𝜑 → ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2) ∧ ((𝐴 gcd 𝐵) = 1 ∧ ¬ 2 ∥ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 (class class class)co 7430 1c1 11153 + caddc 11155 ℕcn 12263 2c2 12318 ↑cexp 14098 ∥ cdvds 16286 gcd cgcd 16527 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-dvds 16287 df-gcd 16528 |
This theorem is referenced by: flt4lem3 42634 flt4lem5a 42638 flt4lem5b 42639 flt4lem5c 42640 flt4lem5d 42641 flt4lem5e 42642 |
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