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Mirrors > Home > MPE Home > Th. List > Mathboxes > flt4lem3 | Structured version Visualization version GIF version |
Description: Equivalent to pythagtriplem4 16787. 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 12615 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) |
3 | flt4lem3.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
4 | 3 | nnzd 12615 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
5 | 2, 4 | zaddcld 12700 | . . 3 ⊢ (𝜑 → (𝐶 + 𝐴) ∈ ℤ) |
6 | 2, 4 | zsubcld 12701 | . . 3 ⊢ (𝜑 → (𝐶 − 𝐴) ∈ ℤ) |
7 | 5, 6 | gcdcomd 16488 | . 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 42071 | . . . 4 ⊢ (𝜑 → ¬ 2 ∥ 𝐵) |
13 | 2nn0 12519 | . . . . . 6 ⊢ 2 ∈ ℕ0 | |
14 | 13 | a1i 11 | . . . . 5 ⊢ (𝜑 → 2 ∈ ℕ0) |
15 | 3, 8, 1, 10, 11 | fltabcoprm 42066 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐵) = 1) |
16 | 3, 8, 1, 14, 11, 15 | fltbccoprm 42065 | . . . 4 ⊢ (𝜑 → (𝐵 gcd 𝐶) = 1) |
17 | 8 | nnsqcld 14238 | . . . . . . 7 ⊢ (𝜑 → (𝐵↑2) ∈ ℕ) |
18 | 17 | nncnd 12258 | . . . . . 6 ⊢ (𝜑 → (𝐵↑2) ∈ ℂ) |
19 | 3 | nnsqcld 14238 | . . . . . . 7 ⊢ (𝜑 → (𝐴↑2) ∈ ℕ) |
20 | 19 | nncnd 12258 | . . . . . 6 ⊢ (𝜑 → (𝐴↑2) ∈ ℂ) |
21 | 18, 20 | addcomd 11446 | . . . . 5 ⊢ (𝜑 → ((𝐵↑2) + (𝐴↑2)) = ((𝐴↑2) + (𝐵↑2))) |
22 | 21, 11 | eqtrd 2768 | . . . 4 ⊢ (𝜑 → ((𝐵↑2) + (𝐴↑2)) = (𝐶↑2)) |
23 | 8, 3, 1, 12, 16, 22 | flt4lem1 42070 | . . 3 ⊢ (𝜑 → ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ ∧ 𝐶 ∈ ℕ) ∧ ((𝐵↑2) + (𝐴↑2)) = (𝐶↑2) ∧ ((𝐵 gcd 𝐴) = 1 ∧ ¬ 2 ∥ 𝐵))) |
24 | pythagtriplem4 16787 | . . 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 2768 | 1 ⊢ (𝜑 → ((𝐶 + 𝐴) gcd (𝐶 − 𝐴)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 (class class class)co 7420 1c1 11139 + caddc 11141 − cmin 11474 ℕcn 12242 2c2 12297 ℕ0cn0 12502 ↑cexp 14058 ∥ cdvds 16230 gcd cgcd 16468 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8286 df-wrecs 8317 df-recs 8391 df-rdg 8430 df-1o 8486 df-2o 8487 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-sup 9465 df-inf 9466 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-n0 12503 df-z 12589 df-uz 12853 df-rp 13007 df-fz 13517 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-dvds 16231 df-gcd 16469 df-prm 16642 |
This theorem is referenced by: (None) |
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