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Mirrors > Home > MPE Home > Th. List > Mathboxes > fltdiv | Structured version Visualization version GIF version |
Description: A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (Contributed by SN, 20-Aug-2024.) |
Ref | Expression |
---|---|
fltdiv.s | ⊢ (𝜑 → 𝑆 ∈ ℂ) |
fltdiv.0 | ⊢ (𝜑 → 𝑆 ≠ 0) |
fltdiv.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
fltdiv.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
fltdiv.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
fltdiv.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
fltdiv.1 | ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) |
Ref | Expression |
---|---|
fltdiv | ⊢ (𝜑 → (((𝐴 / 𝑆)↑𝑁) + ((𝐵 / 𝑆)↑𝑁)) = ((𝐶 / 𝑆)↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fltdiv.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | fltdiv.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
3 | 1, 2 | expcld 14143 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
4 | fltdiv.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
5 | 4, 2 | expcld 14143 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑁) ∈ ℂ) |
6 | fltdiv.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℂ) | |
7 | 6, 2 | expcld 14143 | . . . 4 ⊢ (𝜑 → (𝑆↑𝑁) ∈ ℂ) |
8 | fltdiv.0 | . . . . 5 ⊢ (𝜑 → 𝑆 ≠ 0) | |
9 | 2 | nn0zd 12615 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
10 | 6, 8, 9 | expne0d 14149 | . . . 4 ⊢ (𝜑 → (𝑆↑𝑁) ≠ 0) |
11 | 3, 5, 7, 10 | divdird 12059 | . . 3 ⊢ (𝜑 → (((𝐴↑𝑁) + (𝐵↑𝑁)) / (𝑆↑𝑁)) = (((𝐴↑𝑁) / (𝑆↑𝑁)) + ((𝐵↑𝑁) / (𝑆↑𝑁)))) |
12 | fltdiv.1 | . . . 4 ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) | |
13 | 12 | oveq1d 7435 | . . 3 ⊢ (𝜑 → (((𝐴↑𝑁) + (𝐵↑𝑁)) / (𝑆↑𝑁)) = ((𝐶↑𝑁) / (𝑆↑𝑁))) |
14 | 11, 13 | eqtr3d 2770 | . 2 ⊢ (𝜑 → (((𝐴↑𝑁) / (𝑆↑𝑁)) + ((𝐵↑𝑁) / (𝑆↑𝑁))) = ((𝐶↑𝑁) / (𝑆↑𝑁))) |
15 | 1, 6, 8, 2 | expdivd 14157 | . . 3 ⊢ (𝜑 → ((𝐴 / 𝑆)↑𝑁) = ((𝐴↑𝑁) / (𝑆↑𝑁))) |
16 | 4, 6, 8, 2 | expdivd 14157 | . . 3 ⊢ (𝜑 → ((𝐵 / 𝑆)↑𝑁) = ((𝐵↑𝑁) / (𝑆↑𝑁))) |
17 | 15, 16 | oveq12d 7438 | . 2 ⊢ (𝜑 → (((𝐴 / 𝑆)↑𝑁) + ((𝐵 / 𝑆)↑𝑁)) = (((𝐴↑𝑁) / (𝑆↑𝑁)) + ((𝐵↑𝑁) / (𝑆↑𝑁)))) |
18 | fltdiv.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
19 | 18, 6, 8, 2 | expdivd 14157 | . 2 ⊢ (𝜑 → ((𝐶 / 𝑆)↑𝑁) = ((𝐶↑𝑁) / (𝑆↑𝑁))) |
20 | 14, 17, 19 | 3eqtr4d 2778 | 1 ⊢ (𝜑 → (((𝐴 / 𝑆)↑𝑁) + ((𝐵 / 𝑆)↑𝑁)) = ((𝐶 / 𝑆)↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 (class class class)co 7420 ℂcc 11137 0cc0 11139 + caddc 11142 / cdiv 11902 ℕ0cn0 12503 ↑cexp 14059 |
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 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 |
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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-n0 12504 df-z 12590 df-uz 12854 df-seq 14000 df-exp 14060 |
This theorem is referenced by: fltabcoprmex 42063 |
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