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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 14824 | . . . . 5 ⊢ ((𝐾 ∈ ℤ ∧ 𝐾 ≠ 0 ∧ 𝑀 ∈ ℤ) → (𝐾 ∥ 𝑀 ↔ (𝑀 / 𝐾) ∈ ℤ)) | |
6 | 2, 3, 4, 5 | syl3anc 1318 | . . . 4 ⊢ (𝜑 → (𝐾 ∥ 𝑀 ↔ (𝑀 / 𝐾) ∈ ℤ)) |
7 | 1, 6 | mtbid 313 | . . 3 ⊢ (𝜑 → ¬ (𝑀 / 𝐾) ∈ ℤ) |
8 | df-neg 10148 | . . . . . . 7 ⊢ -𝑁 = (0 − 𝑁) | |
9 | 8 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → -𝑁 = (0 − 𝑁)) |
10 | oveq1 6556 | . . . . . . . 8 ⊢ ((𝑀 + 𝑁) = 0 → ((𝑀 + 𝑁) − 𝑁) = (0 − 𝑁)) | |
11 | 10 | eqcomd 2616 | . . . . . . 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 11359 | . . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℂ) |
14 | etransclem9.n | . . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℤ) | |
15 | 14 | zcnd 11359 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
16 | 13, 15 | pncand 10272 | . . . . . . 7 ⊢ (𝜑 → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
17 | 16 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → ((𝑀 + 𝑁) − 𝑁) = 𝑀) |
18 | 9, 12, 17 | 3eqtrrd 2649 | . . . . 5 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → 𝑀 = -𝑁) |
19 | 18 | oveq1d 6564 | . . . 4 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (𝑀 / 𝐾) = (-𝑁 / 𝐾)) |
20 | etransclem9.kn | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∥ 𝑁) | |
21 | dvdsnegb 14837 | . . . . . . . 8 ⊢ ((𝐾 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝐾 ∥ 𝑁 ↔ 𝐾 ∥ -𝑁)) | |
22 | 2, 14, 21 | syl2anc 691 | . . . . . . 7 ⊢ (𝜑 → (𝐾 ∥ 𝑁 ↔ 𝐾 ∥ -𝑁)) |
23 | 20, 22 | mpbid 221 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∥ -𝑁) |
24 | 14 | znegcld 11360 | . . . . . . 7 ⊢ (𝜑 → -𝑁 ∈ ℤ) |
25 | dvdsval2 14824 | . . . . . . 7 ⊢ ((𝐾 ∈ ℤ ∧ 𝐾 ≠ 0 ∧ -𝑁 ∈ ℤ) → (𝐾 ∥ -𝑁 ↔ (-𝑁 / 𝐾) ∈ ℤ)) | |
26 | 2, 3, 24, 25 | syl3anc 1318 | . . . . . 6 ⊢ (𝜑 → (𝐾 ∥ -𝑁 ↔ (-𝑁 / 𝐾) ∈ ℤ)) |
27 | 23, 26 | mpbid 221 | . . . . 5 ⊢ (𝜑 → (-𝑁 / 𝐾) ∈ ℤ) |
28 | 27 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (-𝑁 / 𝐾) ∈ ℤ) |
29 | 19, 28 | eqeltrd 2688 | . . 3 ⊢ ((𝜑 ∧ (𝑀 + 𝑁) = 0) → (𝑀 / 𝐾) ∈ ℤ) |
30 | 7, 29 | mtand 689 | . 2 ⊢ (𝜑 → ¬ (𝑀 + 𝑁) = 0) |
31 | 30 | neqned 2789 | 1 ⊢ (𝜑 → (𝑀 + 𝑁) ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 (class class class)co 6549 0cc0 9815 + caddc 9818 − cmin 10145 -cneg 10146 / cdiv 10563 ℤcz 11254 ∥ cdvds 14821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-z 11255 df-dvds 14822 |
This theorem is referenced by: etransclem44 39171 |
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