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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 ๐ โ โ0, so the label is more about the context in which this theorem is used). (Contributed by SN, 20-Aug-2024.) |
Ref | Expression |
---|---|
fltmul.s | โข (๐ โ ๐ โ โ) |
fltmul.a | โข (๐ โ ๐ด โ โ) |
fltmul.b | โข (๐ โ ๐ต โ โ) |
fltmul.c | โข (๐ โ ๐ถ โ โ) |
fltmul.n | โข (๐ โ ๐ โ โ0) |
fltmul.1 | โข (๐ โ ((๐ดโ๐) + (๐ตโ๐)) = (๐ถโ๐)) |
Ref | Expression |
---|---|
fltmul | โข (๐ โ (((๐ ยท ๐ด)โ๐) + ((๐ ยท ๐ต)โ๐)) = ((๐ ยท ๐ถ)โ๐)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fltmul.s | . . . . 5 โข (๐ โ ๐ โ โ) | |
2 | fltmul.n | . . . . 5 โข (๐ โ ๐ โ โ0) | |
3 | 1, 2 | expcld 14108 | . . . 4 โข (๐ โ (๐โ๐) โ โ) |
4 | fltmul.a | . . . . 5 โข (๐ โ ๐ด โ โ) | |
5 | 4, 2 | expcld 14108 | . . . 4 โข (๐ โ (๐ดโ๐) โ โ) |
6 | fltmul.b | . . . . 5 โข (๐ โ ๐ต โ โ) | |
7 | 6, 2 | expcld 14108 | . . . 4 โข (๐ โ (๐ตโ๐) โ โ) |
8 | 3, 5, 7 | adddid 11235 | . . 3 โข (๐ โ ((๐โ๐) ยท ((๐ดโ๐) + (๐ตโ๐))) = (((๐โ๐) ยท (๐ดโ๐)) + ((๐โ๐) ยท (๐ตโ๐)))) |
9 | fltmul.1 | . . . 4 โข (๐ โ ((๐ดโ๐) + (๐ตโ๐)) = (๐ถโ๐)) | |
10 | 9 | oveq2d 7417 | . . 3 โข (๐ โ ((๐โ๐) ยท ((๐ดโ๐) + (๐ตโ๐))) = ((๐โ๐) ยท (๐ถโ๐))) |
11 | 8, 10 | eqtr3d 2766 | . 2 โข (๐ โ (((๐โ๐) ยท (๐ดโ๐)) + ((๐โ๐) ยท (๐ตโ๐))) = ((๐โ๐) ยท (๐ถโ๐))) |
12 | 1, 4, 2 | mulexpd 14123 | . . 3 โข (๐ โ ((๐ ยท ๐ด)โ๐) = ((๐โ๐) ยท (๐ดโ๐))) |
13 | 1, 6, 2 | mulexpd 14123 | . . 3 โข (๐ โ ((๐ ยท ๐ต)โ๐) = ((๐โ๐) ยท (๐ตโ๐))) |
14 | 12, 13 | oveq12d 7419 | . 2 โข (๐ โ (((๐ ยท ๐ด)โ๐) + ((๐ ยท ๐ต)โ๐)) = (((๐โ๐) ยท (๐ดโ๐)) + ((๐โ๐) ยท (๐ตโ๐)))) |
15 | fltmul.c | . . 3 โข (๐ โ ๐ถ โ โ) | |
16 | 1, 15, 2 | mulexpd 14123 | . 2 โข (๐ โ ((๐ ยท ๐ถ)โ๐) = ((๐โ๐) ยท (๐ถโ๐))) |
17 | 11, 14, 16 | 3eqtr4d 2774 | 1 โข (๐ โ (((๐ ยท ๐ด)โ๐) + ((๐ ยท ๐ต)โ๐)) = ((๐ ยท ๐ถ)โ๐)) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 (class class class)co 7401 โcc 11104 + caddc 11109 ยท cmul 11111 โ0cn0 12469 โcexp 14024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-seq 13964 df-exp 14025 |
This theorem is referenced by: (None) |
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