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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 14111 | . . . 4 โข (๐ โ (๐โ๐) โ โ) |
4 | fltmul.a | . . . . 5 โข (๐ โ ๐ด โ โ) | |
5 | 4, 2 | expcld 14111 | . . . 4 โข (๐ โ (๐ดโ๐) โ โ) |
6 | fltmul.b | . . . . 5 โข (๐ โ ๐ต โ โ) | |
7 | 6, 2 | expcld 14111 | . . . 4 โข (๐ โ (๐ตโ๐) โ โ) |
8 | 3, 5, 7 | adddid 11238 | . . 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 7425 | . . 3 โข (๐ โ ((๐โ๐) ยท ((๐ดโ๐) + (๐ตโ๐))) = ((๐โ๐) ยท (๐ถโ๐))) |
11 | 8, 10 | eqtr3d 2775 | . 2 โข (๐ โ (((๐โ๐) ยท (๐ดโ๐)) + ((๐โ๐) ยท (๐ตโ๐))) = ((๐โ๐) ยท (๐ถโ๐))) |
12 | 1, 4, 2 | mulexpd 14126 | . . 3 โข (๐ โ ((๐ ยท ๐ด)โ๐) = ((๐โ๐) ยท (๐ดโ๐))) |
13 | 1, 6, 2 | mulexpd 14126 | . . 3 โข (๐ โ ((๐ ยท ๐ต)โ๐) = ((๐โ๐) ยท (๐ตโ๐))) |
14 | 12, 13 | oveq12d 7427 | . 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 14126 | . 2 โข (๐ โ ((๐ ยท ๐ถ)โ๐) = ((๐โ๐) ยท (๐ถโ๐))) |
17 | 11, 14, 16 | 3eqtr4d 2783 | 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 1542 โ wcel 2107 (class class class)co 7409 โcc 11108 + caddc 11113 ยท cmul 11115 โ0cn0 12472 โcexp 14027 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-seq 13967 df-exp 14028 |
This theorem is referenced by: (None) |
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