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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 16219 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝐾 ≠ 0 ∧ 𝑀 ∈ ℤ) → (𝐾 ∥ 𝑀 ↔ (𝑀 / 𝐾) ∈ ℤ)) | |
6 | 2, 3, 4, 5 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → (𝐾 ∥ 𝑀 ↔ (𝑀 / 𝐾) ∈ ℤ)) |
7 | 1, 6 | mtbid 324 | . . 3 ⊢ (𝜑 → ¬ (𝑀 / 𝐾) ∈ ℤ) |
8 | df-neg 11463 | . . . . . . 7 ⊢ -𝑁 = (0 − 𝑁) | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → -𝑁 = (0 − 𝑁)) |
10 | oveq1 7421 | . . . . . . . 8 ⊢ ((𝑀 + 𝑁) = 0 → ((𝑀 + 𝑁) − 𝑁) = (0 − 𝑁)) | |
11 | 10 | eqcomd 2733 | . . . . . . 7 ⊢ ((𝑀 + 𝑁) = 0 → (0 − 𝑁) = ((𝑀 + 𝑁) − 𝑁)) |
12 | 11 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (0 − 𝑁) = ((𝑀 + 𝑁) − 𝑁)) |
13 | 4 | zcnd 12683 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
14 | etransclem9.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
15 | 14 | zcnd 12683 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
16 | 13, 15 | pncand 11588 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
17 | 16 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
18 | 9, 12, 17 | 3eqtrrd 2772 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → 𝑀 = -𝑁) |
19 | 18 | oveq1d 7429 | . . . 4 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (𝑀 / 𝐾) = (-𝑁 / 𝐾)) |
20 | etransclem9.kn | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∥ 𝑁) | |
21 | dvdsnegb 16236 | . . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ 𝑁 ↔ 𝐾 ∥ -𝑁)) | |
22 | 2, 14, 21 | syl2anc 583 | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∥ 𝑁 ↔ 𝐾 ∥ -𝑁)) |
23 | 20, 22 | mpbid 231 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∥ -𝑁) |
24 | 14 | znegcld 12684 | . . . . . . 7 ⊢ (𝜑 → -𝑁 ∈ ℤ) |
25 | dvdsval2 16219 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝐾 ≠ 0 ∧ -𝑁 ∈ ℤ) → (𝐾 ∥ -𝑁 ↔ (-𝑁 / 𝐾) ∈ ℤ)) | |
26 | 2, 3, 24, 25 | syl3anc 1369 | . . . . . 6 ⊢ (𝜑 → (𝐾 ∥ -𝑁 ↔ (-𝑁 / 𝐾) ∈ ℤ)) |
27 | 23, 26 | mpbid 231 | . . . . 5 ⊢ (𝜑 → (-𝑁 / 𝐾) ∈ ℤ) |
28 | 27 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (-𝑁 / 𝐾) ∈ ℤ) |
29 | 19, 28 | eqeltrd 2828 | . . 3 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (𝑀 / 𝐾) ∈ ℤ) |
30 | 7, 29 | mtand 815 | . 2 ⊢ (𝜑 → ¬ (𝑀 + 𝑁) = 0) |
31 | 30 | neqned 2942 | 1 ⊢ (𝜑 → (𝑀 + 𝑁) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2935 class class class wbr 5142 (class class class)co 7414 0cc0 11124 + caddc 11127 − cmin 11460 -cneg 11461 / cdiv 11887 ℤcz 12574 ∥ cdvds 16216 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 |
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 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 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 5143 df-opab 5205 df-mpt 5226 df-tr 5260 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-er 8716 df-en 8954 df-dom 8955 df-sdom 8956 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-z 12575 df-dvds 16217 |
This theorem is referenced by: etransclem44 45579 |
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