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Mirrors > Home > MPE Home > Th. List > Mathboxes > flt4lem3 | Structured version Visualization version GIF version |
Description: Equivalent to pythagtriplem4 16853. Show that 𝐶 + 𝐴 and 𝐶 − 𝐴 are coprime. (Contributed by SN, 22-Aug-2024.) |
Ref | Expression |
---|---|
flt4lem3.a | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
flt4lem3.b | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
flt4lem3.c | ⊢ (𝜑 → 𝐶 ∈ ℕ) |
flt4lem3.1 | ⊢ (𝜑 → 2 ∥ 𝐴) |
flt4lem3.2 | ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) |
flt4lem3.3 | ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) |
Ref | Expression |
---|---|
flt4lem3 | ⊢ (𝜑 → ((𝐶 + 𝐴) gcd (𝐶 − 𝐴)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flt4lem3.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
2 | 1 | nnzd 12638 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) |
3 | flt4lem3.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
4 | 3 | nnzd 12638 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
5 | 2, 4 | zaddcld 12724 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐴) ∈ ℤ) |
6 | 2, 4 | zsubcld 12725 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℤ) |
7 | 5, 6 | gcdcomd 16548 | . 2 ⊢ (𝜑 → ((𝐶 + 𝐴) gcd (𝐶 − 𝐴)) = ((𝐶 − 𝐴) gcd (𝐶 + 𝐴))) |
8 | flt4lem3.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
9 | flt4lem3.1 | . . . . 5 ⊢ (𝜑 → 2 ∥ 𝐴) | |
10 | flt4lem3.2 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐶) = 1) | |
11 | flt4lem3.3 | . . . . 5 ⊢ (𝜑 → ((𝐴↑2) + (𝐵↑2)) = (𝐶↑2)) | |
12 | 3, 8, 1, 9, 10, 11 | flt4lem2 42634 | . . . 4 ⊢ (𝜑 → ¬ 2 ∥ 𝐵) |
13 | 2nn0 12541 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℕ0) |
15 | 3, 8, 1, 10, 11 | fltabcoprm 42629 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) |
16 | 3, 8, 1, 14, 11, 15 | fltbccoprm 42628 | . . . 4 ⊢ (𝜑 → (𝐵 gcd 𝐶) = 1) |
17 | 8 | nnsqcld 14280 | . . . . . . 7 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
18 | 17 | nncnd 12280 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
19 | 3 | nnsqcld 14280 | . . . . . . 7 ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
20 | 19 | nncnd 12280 | . . . . . 6 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
21 | 18, 20 | addcomd 11461 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) + (𝐴↑2)) = ((𝐴↑2) + (𝐵↑2))) |
22 | 21, 11 | eqtrd 2775 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) + (𝐴↑2)) = (𝐶↑2)) |
23 | 8, 3, 1, 12, 16, 22 | flt4lem1 42633 | . . 3 ⊢ (𝜑 → ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐵↑2) + (𝐴↑2)) = (𝐶↑2) ∧ ((𝐵 gcd 𝐴) = 1 ∧ ¬ 2 ∥ 𝐵))) |
24 | pythagtriplem4 16853 | . . 3 ⊢ (((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐵↑2) + (𝐴↑2)) = (𝐶↑2) ∧ ((𝐵 gcd 𝐴) = 1 ∧ ¬ 2 ∥ 𝐵)) → ((𝐶 − 𝐴) gcd (𝐶 + 𝐴)) = 1) | |
25 | 23, 24 | syl 17 | . 2 ⊢ (𝜑 → ((𝐶 − 𝐴) gcd (𝐶 + 𝐴)) = 1) |
26 | 7, 25 | eqtrd 2775 | 1 ⊢ (𝜑 → ((𝐶 + 𝐴) gcd (𝐶 − 𝐴)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 (class class class)co 7431 1c1 11154 + caddc 11156 − cmin 11490 ℕcn 12264 2c2 12319 ℕ0cn0 12524 ↑cexp 14099 ∥ cdvds 16287 gcd cgcd 16528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-n0 12525 df-z 12612 df-uz 12877 df-rp 13033 df-fz 13545 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-dvds 16288 df-gcd 16529 df-prm 16706 |
This theorem is referenced by: (None) |
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