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Mirrors > Home > MPE Home > Th. List > Mathboxes > fltmul | 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. (There does not seem to be a standard term for Fermat or Pythagorean triples extended to any 𝑁 ∈ ℕ0, hence the label is more about the context in which this theorem is used). (Contributed by SN, 20-Aug-2024.) |
Ref | Expression |
---|---|
fltmul.s | ⊢ (𝜑 → 𝑆 ∈ ℂ) |
fltmul.a | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
fltmul.b | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
fltmul.c | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
fltmul.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
fltmul.1 | ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) |
Ref | Expression |
---|---|
fltmul | ⊢ (𝜑 → (((𝑆 · 𝐴)↑𝑁) + ((𝑆 · 𝐵)↑𝑁)) = ((𝑆 · 𝐶)↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fltmul.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ ℂ) | |
2 | fltmul.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
3 | 1, 2 | expcld 13602 | . . . 4 ⊢ (𝜑 → (𝑆↑𝑁) ∈ ℂ) |
4 | fltmul.a | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
5 | 4, 2 | expcld 13602 | . . . 4 ⊢ (𝜑 → (𝐴↑𝑁) ∈ ℂ) |
6 | fltmul.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
7 | 6, 2 | expcld 13602 | . . . 4 ⊢ (𝜑 → (𝐵↑𝑁) ∈ ℂ) |
8 | 3, 5, 7 | adddid 10743 | . . 3 ⊢ (𝜑 → ((𝑆↑𝑁) · ((𝐴↑𝑁) + (𝐵↑𝑁))) = (((𝑆↑𝑁) · (𝐴↑𝑁)) + ((𝑆↑𝑁) · (𝐵↑𝑁)))) |
9 | fltmul.1 | . . . 4 ⊢ (𝜑 → ((𝐴↑𝑁) + (𝐵↑𝑁)) = (𝐶↑𝑁)) | |
10 | 9 | oveq2d 7186 | . . 3 ⊢ (𝜑 → ((𝑆↑𝑁) · ((𝐴↑𝑁) + (𝐵↑𝑁))) = ((𝑆↑𝑁) · (𝐶↑𝑁))) |
11 | 8, 10 | eqtr3d 2775 | . 2 ⊢ (𝜑 → (((𝑆↑𝑁) · (𝐴↑𝑁)) + ((𝑆↑𝑁) · (𝐵↑𝑁))) = ((𝑆↑𝑁) · (𝐶↑𝑁))) |
12 | 1, 4, 2 | mulexpd 13617 | . . 3 ⊢ (𝜑 → ((𝑆 · 𝐴)↑𝑁) = ((𝑆↑𝑁) · (𝐴↑𝑁))) |
13 | 1, 6, 2 | mulexpd 13617 | . . 3 ⊢ (𝜑 → ((𝑆 · 𝐵)↑𝑁) = ((𝑆↑𝑁) · (𝐵↑𝑁))) |
14 | 12, 13 | oveq12d 7188 | . 2 ⊢ (𝜑 → (((𝑆 · 𝐴)↑𝑁) + ((𝑆 · 𝐵)↑𝑁)) = (((𝑆↑𝑁) · (𝐴↑𝑁)) + ((𝑆↑𝑁) · (𝐵↑𝑁)))) |
15 | fltmul.c | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
16 | 1, 15, 2 | mulexpd 13617 | . 2 ⊢ (𝜑 → ((𝑆 · 𝐶)↑𝑁) = ((𝑆↑𝑁) · (𝐶↑𝑁))) |
17 | 11, 14, 16 | 3eqtr4d 2783 | 1 ⊢ (𝜑 → (((𝑆 · 𝐴)↑𝑁) + ((𝑆 · 𝐵)↑𝑁)) = ((𝑆 · 𝐶)↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 (class class class)co 7170 ℂcc 10613 + caddc 10618 · cmul 10620 ℕ0cn0 11976 ↑cexp 13521 |
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 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
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 2075 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 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-n0 11977 df-z 12063 df-uz 12325 df-seq 13461 df-exp 13522 |
This theorem is referenced by: (None) |
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