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Mirrors > Home > MPE Home > Th. List > Mathboxes > fltdvdsabdvdsc | Structured version Visualization version GIF version |
Description: Any factor of both 𝐴 and 𝐵 also divides 𝐶. This establishes the validity of fltabcoprmex 41870. (Contributed by SN, 21-Aug-2024.) |
Ref | Expression |
---|---|
fltdvdsabdvdsc.a | ⊢ (𝜑 → 𝐴 ∈ ℕ) |
fltdvdsabdvdsc.b | ⊢ (𝜑 → 𝐵 ∈ ℕ) |
fltdvdsabdvdsc.c | ⊢ (𝜑 → 𝐶 ∈ ℕ) |
fltdvdsabdvdsc.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
fltdvdsabdvdsc.1 | ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) |
Ref | Expression |
---|---|
fltdvdsabdvdsc | ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fltdvdsabdvdsc.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℕ) | |
2 | fltdvdsabdvdsc.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℕ) | |
3 | gcdnncl 16445 | . . . . . . 7 ⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 gcd 𝐵) ∈ ℕ) | |
4 | 1, 2, 3 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℕ) |
5 | fltdvdsabdvdsc.n | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
6 | 5 | nnnn0d 12529 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
7 | 4, 6 | nnexpcld 14205 | . . . . 5 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∈ ℕ) |
8 | 7 | nnzd 12582 | . . . 4 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∈ ℤ) |
9 | 1, 6 | nnexpcld 14205 | . . . . 5 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℕ) |
10 | 9 | nnzd 12582 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℤ) |
11 | 2, 6 | nnexpcld 14205 | . . . . 5 ⊢ (𝜑 → (𝐵↑𝑁) ∈ ℕ) |
12 | 11 | nnzd 12582 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑁) ∈ ℤ) |
13 | 4 | nnzd 12582 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∈ ℤ) |
14 | 1 | nnzd 12582 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℤ) |
15 | 2 | nnzd 12582 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℤ) |
16 | gcddvds 16441 | . . . . . . 7 ⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵)) | |
17 | 14, 15, 16 | syl2anc 583 | . . . . . 6 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐴 ∧ (𝐴 gcd 𝐵) ∥ 𝐵)) |
18 | 17 | simpld 494 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐴) |
19 | 13, 14, 6, 18 | dvdsexpad 41712 | . . . 4 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∥ (𝐴↑𝑁)) |
20 | 17 | simprd 495 | . . . . 5 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐵) |
21 | 13, 15, 6, 20 | dvdsexpad 41712 | . . . 4 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∥ (𝐵↑𝑁)) |
22 | 8, 10, 12, 19, 21 | dvds2addd 16232 | . . 3 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∥ ((𝐴↑𝑁) + (𝐵↑𝑁))) |
23 | fltdvdsabdvdsc.1 | . . 3 ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) | |
24 | 22, 23 | breqtrd 5164 | . 2 ⊢ (𝜑 → ((𝐴 gcd 𝐵)↑𝑁) ∥ (𝐶↑𝑁)) |
25 | fltdvdsabdvdsc.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℕ) | |
26 | dvdsexpnn 41720 | . . 3 ⊢ (((𝐴 gcd 𝐵) ∈ ℕ ∧ 𝐶 ∈ ℕ ∧ 𝑁 ∈ ℕ) → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑𝑁) ∥ (𝐶↑𝑁))) | |
27 | 4, 25, 5, 26 | syl3anc 1368 | . 2 ⊢ (𝜑 → ((𝐴 gcd 𝐵) ∥ 𝐶 ↔ ((𝐴 gcd 𝐵)↑𝑁) ∥ (𝐶↑𝑁))) |
28 | 24, 27 | mpbird 257 | 1 ⊢ (𝜑 → (𝐴 gcd 𝐵) ∥ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 class class class wbr 5138 (class class class)co 7401 + caddc 11109 ℕcn 12209 ℤcz 12555 ↑cexp 14024 ∥ cdvds 16194 gcd cgcd 16432 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-sup 9433 df-inf 9434 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-div 11869 df-nn 12210 df-2 12272 df-3 12273 df-n0 12470 df-z 12556 df-uz 12820 df-rp 12972 df-fl 13754 df-mod 13832 df-seq 13964 df-exp 14025 df-cj 15043 df-re 15044 df-im 15045 df-sqrt 15179 df-abs 15180 df-dvds 16195 df-gcd 16433 |
This theorem is referenced by: flt4lem2 41878 |
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