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Mirrors > Home > MPE Home > Th. List > Mathboxes > flt0 | Structured version Visualization version GIF version |
Description: A counterexample for FLT does not exist for 𝑁 = 0. (Contributed by SN, 20-Aug-2024.) |
Ref | Expression |
---|---|
flt0.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
flt0.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
flt0.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
flt0.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
flt0.1 | ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) |
Ref | Expression |
---|---|
flt0 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | flt0.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
2 | 1p1e2 11834 | . . . . . 6 ⊢ (1 + 1) = 2 | |
3 | sn-1ne2 39855 | . . . . . . 7 ⊢ 1 ≠ 2 | |
4 | 3 | necomi 2988 | . . . . . 6 ⊢ 2 ≠ 1 |
5 | 2, 4 | eqnetri 3004 | . . . . 5 ⊢ (1 + 1) ≠ 1 |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → (1 + 1) ≠ 1) |
7 | flt0.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
8 | 7 | exp0d 13589 | . . . . 5 ⊢ (𝜑 → (𝐴↑0) = 1) |
9 | flt0.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
10 | 9 | exp0d 13589 | . . . . 5 ⊢ (𝜑 → (𝐵↑0) = 1) |
11 | 8, 10 | oveq12d 7182 | . . . 4 ⊢ (𝜑 → ((𝐴↑0) + (𝐵↑0)) = (1 + 1)) |
12 | flt0.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
13 | 12 | exp0d 13589 | . . . 4 ⊢ (𝜑 → (𝐶↑0) = 1) |
14 | 6, 11, 13 | 3netr4d 3011 | . . 3 ⊢ (𝜑 → ((𝐴↑0) + (𝐵↑0)) ≠ (𝐶↑0)) |
15 | flt0.1 | . . . . 5 ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) | |
16 | oveq2 7172 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝐴↑𝑁) = (𝐴↑0)) | |
17 | oveq2 7172 | . . . . . . 7 ⊢ (𝑁 = 0 → (𝐵↑𝑁) = (𝐵↑0)) | |
18 | 16, 17 | oveq12d 7182 | . . . . . 6 ⊢ (𝑁 = 0 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = ((𝐴↑0) + (𝐵↑0))) |
19 | oveq2 7172 | . . . . . 6 ⊢ (𝑁 = 0 → (𝐶↑𝑁) = (𝐶↑0)) | |
20 | 18, 19 | eqeq12d 2754 | . . . . 5 ⊢ (𝑁 = 0 → (((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁) ↔ ((𝐴↑0) + (𝐵↑0)) = (𝐶↑0))) |
21 | 15, 20 | syl5ibcom 248 | . . . 4 ⊢ (𝜑 → (𝑁 = 0 → ((𝐴↑0) + (𝐵↑0)) = (𝐶↑0))) |
22 | 21 | imp 410 | . . 3 ⊢ ((𝜑 ∧ 𝑁 = 0) → ((𝐴↑0) + (𝐵↑0)) = (𝐶↑0)) |
23 | 14, 22 | mteqand 3037 | . 2 ⊢ (𝜑 → 𝑁 ≠ 0) |
24 | elnnne0 11983 | . 2 ⊢ (𝑁 ∈ ℕ ↔ (𝑁 ∈ ℕ0 ∧ 𝑁 ≠ 0)) | |
25 | 1, 23, 24 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2113 ≠ wne 2934 (class class class)co 7164 ℂcc 10606 0cc0 10608 1c1 10609 + caddc 10611 ℕcn 11709 2c2 11764 ℕ0cn0 11969 ↑cexp 13514 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-resscn 10665 ax-1cn 10666 ax-icn 10667 ax-addcl 10668 ax-addrcl 10669 ax-mulcl 10670 ax-mulrcl 10671 ax-mulcom 10672 ax-addass 10673 ax-mulass 10674 ax-distr 10675 ax-i2m1 10676 ax-1ne0 10677 ax-1rid 10678 ax-rnegex 10679 ax-rrecex 10680 ax-cnre 10681 ax-pre-lttri 10682 ax-pre-lttrn 10683 ax-pre-ltadd 10684 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-er 8313 df-en 8549 df-dom 8550 df-sdom 8551 df-pnf 10748 df-mnf 10749 df-xr 10750 df-ltxr 10751 df-le 10752 df-neg 10944 df-nn 11710 df-2 11772 df-n0 11970 df-z 12056 df-seq 13454 df-exp 13515 |
This theorem is referenced by: fltaccoprm 40033 |
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