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Mirrors > Home > MPE Home > Th. List > gcdabsOLD | Structured version Visualization version GIF version |
Description: Obsolete version of gcdabs 16503 as of 15-Sep-2024. (Contributed by Paul Chapman, 31-Mar-2011.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
gcdabsOLD | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | zre 12590 | . . 3 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ) | |
2 | zre 12590 | . . 3 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
3 | absor 15277 | . . . 4 ⊢ (𝑀 ∈ ℝ → ((abs‘𝑀) = 𝑀 ∨ (abs‘𝑀) = -𝑀)) | |
4 | absor 15277 | . . . 4 ⊢ (𝑁 ∈ ℝ → ((abs‘𝑁) = 𝑁 ∨ (abs‘𝑁) = -𝑁)) | |
5 | 3, 4 | anim12i 611 | . . 3 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (((abs‘𝑀) = 𝑀 ∨ (abs‘𝑀) = -𝑀) ∧ ((abs‘𝑁) = 𝑁 ∨ (abs‘𝑁) = -𝑁))) |
6 | 1, 2, 5 | syl2an 594 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) = 𝑀 ∨ (abs‘𝑀) = -𝑀) ∧ ((abs‘𝑁) = 𝑁 ∨ (abs‘𝑁) = -𝑁))) |
7 | oveq12 7424 | . . . 4 ⊢ (((abs‘𝑀) = 𝑀 ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁)) | |
8 | 7 | a1i 11 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) = 𝑀 ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))) |
9 | oveq12 7424 | . . . . 5 ⊢ (((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑀) gcd (abs‘𝑁)) = (-𝑀 gcd 𝑁)) | |
10 | neggcd 16495 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑀 gcd 𝑁) = (𝑀 gcd 𝑁)) | |
11 | 9, 10 | sylan9eqr 2787 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = 𝑁)) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁)) |
12 | 11 | ex 411 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = 𝑁) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))) |
13 | oveq12 7424 | . . . . 5 ⊢ (((abs‘𝑀) = 𝑀 ∧ (abs‘𝑁) = -𝑁) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd -𝑁)) | |
14 | gcdneg 16494 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 gcd -𝑁) = (𝑀 gcd 𝑁)) | |
15 | 13, 14 | sylan9eqr 2787 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((abs‘𝑀) = 𝑀 ∧ (abs‘𝑁) = -𝑁)) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁)) |
16 | 15 | ex 411 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) = 𝑀 ∧ (abs‘𝑁) = -𝑁) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))) |
17 | oveq12 7424 | . . . . 5 ⊢ (((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = -𝑁) → ((abs‘𝑀) gcd (abs‘𝑁)) = (-𝑀 gcd -𝑁)) | |
18 | znegcl 12625 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → -𝑀 ∈ ℤ) | |
19 | gcdneg 16494 | . . . . . . 7 ⊢ ((-𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑀 gcd -𝑁) = (-𝑀 gcd 𝑁)) | |
20 | 18, 19 | sylan 578 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑀 gcd -𝑁) = (-𝑀 gcd 𝑁)) |
21 | 20, 10 | eqtrd 2765 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑀 gcd -𝑁) = (𝑀 gcd 𝑁)) |
22 | 17, 21 | sylan9eqr 2787 | . . . 4 ⊢ (((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = -𝑁)) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁)) |
23 | 22 | ex 411 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (((abs‘𝑀) = -𝑀 ∧ (abs‘𝑁) = -𝑁) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))) |
24 | 8, 12, 16, 23 | ccased 1036 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((((abs‘𝑀) = 𝑀 ∨ (abs‘𝑀) = -𝑀) ∧ ((abs‘𝑁) = 𝑁 ∨ (abs‘𝑁) = -𝑁)) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁))) |
25 | 6, 24 | mpd 15 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((abs‘𝑀) gcd (abs‘𝑁)) = (𝑀 gcd 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 845 = wceq 1533 ∈ wcel 2098 ‘cfv 6542 (class class class)co 7415 ℝcr 11135 -cneg 11473 ℤcz 12586 abscabs 15211 gcd cgcd 16466 |
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 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 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 5144 df-opab 5206 df-mpt 5227 df-tr 5261 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 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 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 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-er 8721 df-en 8961 df-dom 8962 df-sdom 8963 df-sup 9463 df-inf 9464 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-n0 12501 df-z 12587 df-uz 12851 df-rp 13005 df-seq 13997 df-exp 14057 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-dvds 16229 df-gcd 16467 |
This theorem is referenced by: (None) |
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