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Mirrors > Home > ILE Home > Th. List > subsq2 | GIF version |
Description: Express the difference of the squares of two numbers as a polynomial in the difference of the numbers. (Contributed by NM, 21-Feb-2008.) |
Ref | Expression |
---|---|
subsq2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = (((𝐴 − 𝐵)↑2) + ((2 · 𝐵) · (𝐴 − 𝐵)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2cn 8949 | . . . . . . . 8 ⊢ 2 ∈ ℂ | |
2 | mulcl 7901 | . . . . . . . 8 ⊢ ((2 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐵) ∈ ℂ) | |
3 | 1, 2 | mpan 422 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → (2 · 𝐵) ∈ ℂ) |
4 | 3 | adantl 275 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · 𝐵) ∈ ℂ) |
5 | subadd23 8131 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (2 · 𝐵) ∈ ℂ) → ((𝐴 − 𝐵) + (2 · 𝐵)) = (𝐴 + ((2 · 𝐵) − 𝐵))) | |
6 | 4, 5 | mpd3an3 1333 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + (2 · 𝐵)) = (𝐴 + ((2 · 𝐵) − 𝐵))) |
7 | 2times 9006 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℂ → (2 · 𝐵) = (𝐵 + 𝐵)) | |
8 | 7 | oveq1d 5868 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → ((2 · 𝐵) − 𝐵) = ((𝐵 + 𝐵) − 𝐵)) |
9 | pncan 8125 | . . . . . . . . 9 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐵 + 𝐵) − 𝐵) = 𝐵) | |
10 | 9 | anidms 395 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → ((𝐵 + 𝐵) − 𝐵) = 𝐵) |
11 | 8, 10 | eqtrd 2203 | . . . . . . 7 ⊢ (𝐵 ∈ ℂ → ((2 · 𝐵) − 𝐵) = 𝐵) |
12 | 11 | adantl 275 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((2 · 𝐵) − 𝐵) = 𝐵) |
13 | 12 | oveq2d 5869 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + ((2 · 𝐵) − 𝐵)) = (𝐴 + 𝐵)) |
14 | 6, 13 | eqtrd 2203 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) + (2 · 𝐵)) = (𝐴 + 𝐵)) |
15 | 14 | oveq1d 5868 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 − 𝐵) + (2 · 𝐵)) · (𝐴 − 𝐵)) = ((𝐴 + 𝐵) · (𝐴 − 𝐵))) |
16 | subcl 8118 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
17 | 16, 4, 16 | adddird 7945 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 − 𝐵) + (2 · 𝐵)) · (𝐴 − 𝐵)) = (((𝐴 − 𝐵) · (𝐴 − 𝐵)) + ((2 · 𝐵) · (𝐴 − 𝐵)))) |
18 | 15, 17 | eqtr3d 2205 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · (𝐴 − 𝐵)) = (((𝐴 − 𝐵) · (𝐴 − 𝐵)) + ((2 · 𝐵) · (𝐴 − 𝐵)))) |
19 | subsq 10582 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = ((𝐴 + 𝐵) · (𝐴 − 𝐵))) | |
20 | sqval 10534 | . . . 4 ⊢ ((𝐴 − 𝐵) ∈ ℂ → ((𝐴 − 𝐵)↑2) = ((𝐴 − 𝐵) · (𝐴 − 𝐵))) | |
21 | 16, 20 | syl 14 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵)↑2) = ((𝐴 − 𝐵) · (𝐴 − 𝐵))) |
22 | 21 | oveq1d 5868 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 − 𝐵)↑2) + ((2 · 𝐵) · (𝐴 − 𝐵))) = (((𝐴 − 𝐵) · (𝐴 − 𝐵)) + ((2 · 𝐵) · (𝐴 − 𝐵)))) |
23 | 18, 19, 22 | 3eqtr4d 2213 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) − (𝐵↑2)) = (((𝐴 − 𝐵)↑2) + ((2 · 𝐵) · (𝐴 − 𝐵)))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1348 ∈ wcel 2141 (class class class)co 5853 ℂcc 7772 + caddc 7777 · cmul 7779 − cmin 8090 2c2 8929 ↑cexp 10475 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-mulrcl 7873 ax-addcom 7874 ax-mulcom 7875 ax-addass 7876 ax-mulass 7877 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-1rid 7881 ax-0id 7882 ax-rnegex 7883 ax-precex 7884 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890 ax-pre-mulgt0 7891 ax-pre-mulext 7892 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-po 4281 df-iso 4282 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-reap 8494 df-ap 8501 df-div 8590 df-inn 8879 df-2 8937 df-n0 9136 df-z 9213 df-uz 9488 df-seqfrec 10402 df-exp 10476 |
This theorem is referenced by: (None) |
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