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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2itscplem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2itscp 44076. (Contributed by AV, 4-Mar-2023.) |
Ref | Expression |
---|---|
2itscp.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
2itscp.b | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
2itscp.x | ⊢ (𝜑 → 𝑋 ∈ ℝ) |
2itscp.y | ⊢ (𝜑 → 𝑌 ∈ ℝ) |
2itscp.d | ⊢ 𝐷 = (𝑋 − 𝐴) |
2itscp.e | ⊢ 𝐸 = (𝐵 − 𝑌) |
2itscp.c | ⊢ 𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴)) |
Ref | Expression |
---|---|
2itscplem2 | ⊢ (𝜑 → (𝐶↑2) = ((((𝐷↑2) · (𝐵↑2)) + (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) + ((𝐸↑2) · (𝐴↑2)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 2itscp.c | . . . 4 ⊢ 𝐶 = ((𝐷 · 𝐵) + (𝐸 · 𝐴)) | |
2 | 1 | oveq1i 6980 | . . 3 ⊢ (𝐶↑2) = (((𝐷 · 𝐵) + (𝐸 · 𝐴))↑2) |
3 | 2 | a1i 11 | . 2 ⊢ (𝜑 → (𝐶↑2) = (((𝐷 · 𝐵) + (𝐸 · 𝐴))↑2)) |
4 | 2itscp.d | . . . . 5 ⊢ 𝐷 = (𝑋 − 𝐴) | |
5 | 2itscp.x | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ℝ) | |
6 | 5 | recnd 10460 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
7 | 2itscp.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
8 | 7 | recnd 10460 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
9 | 6, 8 | subcld 10790 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝐴) ∈ ℂ) |
10 | 4, 9 | syl5eqel 2864 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
11 | 2itscp.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
12 | 11 | recnd 10460 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
13 | 10, 12 | mulcld 10452 | . . 3 ⊢ (𝜑 → (𝐷 · 𝐵) ∈ ℂ) |
14 | 2itscp.e | . . . . 5 ⊢ 𝐸 = (𝐵 − 𝑌) | |
15 | 2itscp.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
16 | 15 | recnd 10460 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℂ) |
17 | 12, 16 | subcld 10790 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝑌) ∈ ℂ) |
18 | 14, 17 | syl5eqel 2864 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
19 | 18, 8 | mulcld 10452 | . . 3 ⊢ (𝜑 → (𝐸 · 𝐴) ∈ ℂ) |
20 | binom2 13387 | . . 3 ⊢ (((𝐷 · 𝐵) ∈ ℂ ∧ (𝐸 · 𝐴) ∈ ℂ) → (((𝐷 · 𝐵) + (𝐸 · 𝐴))↑2) = ((((𝐷 · 𝐵)↑2) + (2 · ((𝐷 · 𝐵) · (𝐸 · 𝐴)))) + ((𝐸 · 𝐴)↑2))) | |
21 | 13, 19, 20 | syl2anc 576 | . 2 ⊢ (𝜑 → (((𝐷 · 𝐵) + (𝐸 · 𝐴))↑2) = ((((𝐷 · 𝐵)↑2) + (2 · ((𝐷 · 𝐵) · (𝐸 · 𝐴)))) + ((𝐸 · 𝐴)↑2))) |
22 | 10, 12 | sqmuld 13330 | . . . 4 ⊢ (𝜑 → ((𝐷 · 𝐵)↑2) = ((𝐷↑2) · (𝐵↑2))) |
23 | mul4r 10601 | . . . . . 6 ⊢ (((𝐷 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐸 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → ((𝐷 · 𝐵) · (𝐸 · 𝐴)) = ((𝐷 · 𝐴) · (𝐸 · 𝐵))) | |
24 | 10, 12, 18, 8, 23 | syl22anc 826 | . . . . 5 ⊢ (𝜑 → ((𝐷 · 𝐵) · (𝐸 · 𝐴)) = ((𝐷 · 𝐴) · (𝐸 · 𝐵))) |
25 | 24 | oveq2d 6986 | . . . 4 ⊢ (𝜑 → (2 · ((𝐷 · 𝐵) · (𝐸 · 𝐴))) = (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) |
26 | 22, 25 | oveq12d 6988 | . . 3 ⊢ (𝜑 → (((𝐷 · 𝐵)↑2) + (2 · ((𝐷 · 𝐵) · (𝐸 · 𝐴)))) = (((𝐷↑2) · (𝐵↑2)) + (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵))))) |
27 | 18, 8 | sqmuld 13330 | . . 3 ⊢ (𝜑 → ((𝐸 · 𝐴)↑2) = ((𝐸↑2) · (𝐴↑2))) |
28 | 26, 27 | oveq12d 6988 | . 2 ⊢ (𝜑 → ((((𝐷 · 𝐵)↑2) + (2 · ((𝐷 · 𝐵) · (𝐸 · 𝐴)))) + ((𝐸 · 𝐴)↑2)) = ((((𝐷↑2) · (𝐵↑2)) + (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) + ((𝐸↑2) · (𝐴↑2)))) |
29 | 3, 21, 28 | 3eqtrd 2812 | 1 ⊢ (𝜑 → (𝐶↑2) = ((((𝐷↑2) · (𝐵↑2)) + (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) + ((𝐸↑2) · (𝐴↑2)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1507 ∈ wcel 2048 (class class class)co 6970 ℂcc 10325 ℝcr 10326 + caddc 10330 · cmul 10332 − cmin 10662 2c2 11488 ↑cexp 13237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10383 ax-resscn 10384 ax-1cn 10385 ax-icn 10386 ax-addcl 10387 ax-addrcl 10388 ax-mulcl 10389 ax-mulrcl 10390 ax-mulcom 10391 ax-addass 10392 ax-mulass 10393 ax-distr 10394 ax-i2m1 10395 ax-1ne0 10396 ax-1rid 10397 ax-rnegex 10398 ax-rrecex 10399 ax-cnre 10400 ax-pre-lttri 10401 ax-pre-lttrn 10402 ax-pre-ltadd 10403 ax-pre-mulgt0 10404 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-reu 3089 df-rab 3091 df-v 3411 df-sbc 3678 df-csb 3783 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-iun 4788 df-br 4924 df-opab 4986 df-mpt 5003 df-tr 5025 df-id 5305 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-xp 5406 df-rel 5407 df-cnv 5408 df-co 5409 df-dm 5410 df-rn 5411 df-res 5412 df-ima 5413 df-pred 5980 df-ord 6026 df-on 6027 df-lim 6028 df-suc 6029 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-riota 6931 df-ov 6973 df-oprab 6974 df-mpo 6975 df-om 7391 df-2nd 7495 df-wrecs 7743 df-recs 7805 df-rdg 7843 df-er 8081 df-en 8299 df-dom 8300 df-sdom 8301 df-pnf 10468 df-mnf 10469 df-xr 10470 df-ltxr 10471 df-le 10472 df-sub 10664 df-neg 10665 df-nn 11432 df-2 11496 df-n0 11701 df-z 11787 df-uz 12052 df-seq 13178 df-exp 13238 |
This theorem is referenced by: 2itscplem3 44075 |
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