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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > etransclem9 | Structured version Visualization version GIF version |
Description: If 𝐾 divides 𝑁 but 𝐾 does not divide 𝑀 then 𝑀 + 𝑁 cannot be zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
etransclem9.k | ⊢ (𝜑 → 𝐾 ∈ ℤ) |
etransclem9.kn0 | ⊢ (𝜑 → 𝐾 ≠ 0) |
etransclem9.m | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
etransclem9.n | ⊢ (𝜑 → 𝑁 ∈ ℤ) |
etransclem9.km | ⊢ (𝜑 → ¬ 𝐾 ∥ 𝑀) |
etransclem9.kn | ⊢ (𝜑 → 𝐾 ∥ 𝑁) |
Ref | Expression |
---|---|
etransclem9 | ⊢ (𝜑 → (𝑀 + 𝑁) ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | etransclem9.km | . . . 4 ⊢ (𝜑 → ¬ 𝐾 ∥ 𝑀) | |
2 | etransclem9.k | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ ℤ) | |
3 | etransclem9.kn0 | . . . . 5 ⊢ (𝜑 → 𝐾 ≠ 0) | |
4 | etransclem9.m | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
5 | dvdsval2 15390 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝐾 ≠ 0 ∧ 𝑀 ∈ ℤ) → (𝐾 ∥ 𝑀 ↔ (𝑀 / 𝐾) ∈ ℤ)) | |
6 | 2, 3, 4, 5 | syl3anc 1439 | . . . 4 ⊢ (𝜑 → (𝐾 ∥ 𝑀 ↔ (𝑀 / 𝐾) ∈ ℤ)) |
7 | 1, 6 | mtbid 316 | . . 3 ⊢ (𝜑 → ¬ (𝑀 / 𝐾) ∈ ℤ) |
8 | df-neg 10609 | . . . . . . 7 ⊢ -𝑁 = (0 − 𝑁) | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → -𝑁 = (0 − 𝑁)) |
10 | oveq1 6929 | . . . . . . . 8 ⊢ ((𝑀 + 𝑁) = 0 → ((𝑀 + 𝑁) − 𝑁) = (0 − 𝑁)) | |
11 | 10 | eqcomd 2784 | . . . . . . 7 ⊢ ((𝑀 + 𝑁) = 0 → (0 − 𝑁) = ((𝑀 + 𝑁) − 𝑁)) |
12 | 11 | adantl 475 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (0 − 𝑁) = ((𝑀 + 𝑁) − 𝑁)) |
13 | 4 | zcnd 11835 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
14 | etransclem9.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
15 | 14 | zcnd 11835 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
16 | 13, 15 | pncand 10735 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
17 | 16 | adantr 474 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
18 | 9, 12, 17 | 3eqtrrd 2819 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → 𝑀 = -𝑁) |
19 | 18 | oveq1d 6937 | . . . 4 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (𝑀 / 𝐾) = (-𝑁 / 𝐾)) |
20 | etransclem9.kn | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∥ 𝑁) | |
21 | dvdsnegb 15406 | . . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ 𝑁 ↔ 𝐾 ∥ -𝑁)) | |
22 | 2, 14, 21 | syl2anc 579 | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∥ 𝑁 ↔ 𝐾 ∥ -𝑁)) |
23 | 20, 22 | mpbid 224 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∥ -𝑁) |
24 | 14 | znegcld 11836 | . . . . . . 7 ⊢ (𝜑 → -𝑁 ∈ ℤ) |
25 | dvdsval2 15390 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝐾 ≠ 0 ∧ -𝑁 ∈ ℤ) → (𝐾 ∥ -𝑁 ↔ (-𝑁 / 𝐾) ∈ ℤ)) | |
26 | 2, 3, 24, 25 | syl3anc 1439 | . . . . . 6 ⊢ (𝜑 → (𝐾 ∥ -𝑁 ↔ (-𝑁 / 𝐾) ∈ ℤ)) |
27 | 23, 26 | mpbid 224 | . . . . 5 ⊢ (𝜑 → (-𝑁 / 𝐾) ∈ ℤ) |
28 | 27 | adantr 474 | . . . 4 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (-𝑁 / 𝐾) ∈ ℤ) |
29 | 19, 28 | eqeltrd 2859 | . . 3 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (𝑀 / 𝐾) ∈ ℤ) |
30 | 7, 29 | mtand 806 | . 2 ⊢ (𝜑 → ¬ (𝑀 + 𝑁) = 0) |
31 | 30 | neqned 2976 | 1 ⊢ (𝜑 → (𝑀 + 𝑁) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 class class class wbr 4886 (class class class)co 6922 0cc0 10272 + caddc 10275 − cmin 10606 -cneg 10607 / cdiv 11032 ℤcz 11728 ∥ cdvds 15387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-z 11729 df-dvds 15388 |
This theorem is referenced by: etransclem44 41426 |
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