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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2itscplem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2itscp 44842. (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 7159 | . . 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 10662 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
7 | 2itscp.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
8 | 7 | recnd 10662 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
9 | 6, 8 | subcld 10990 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝐴) ∈ ℂ) |
10 | 4, 9 | eqeltrid 2916 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
11 | 2itscp.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
12 | 11 | recnd 10662 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
13 | 10, 12 | mulcld 10654 | . . 3 ⊢ (𝜑 → (𝐷 · 𝐵) ∈ ℂ) |
14 | 2itscp.e | . . . . 5 ⊢ 𝐸 = (𝐵 − 𝑌) | |
15 | 2itscp.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
16 | 15 | recnd 10662 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℂ) |
17 | 12, 16 | subcld 10990 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝑌) ∈ ℂ) |
18 | 14, 17 | eqeltrid 2916 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
19 | 18, 8 | mulcld 10654 | . . 3 ⊢ (𝜑 → (𝐸 · 𝐴) ∈ ℂ) |
20 | binom2 13576 | . . 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 586 | . 2 ⊢ (𝜑 → (((𝐷 · 𝐵) + (𝐸 · 𝐴))↑2) = ((((𝐷 · 𝐵)↑2) + (2 · ((𝐷 · 𝐵) · (𝐸 · 𝐴)))) + ((𝐸 · 𝐴)↑2))) |
22 | 10, 12 | sqmuld 13519 | . . . 4 ⊢ (𝜑 → ((𝐷 · 𝐵)↑2) = ((𝐷↑2) · (𝐵↑2))) |
23 | mul4r 10802 | . . . . . 6 ⊢ (((𝐷 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ (𝐸 ∈ ℂ ∧ 𝐴 ∈ ℂ)) → ((𝐷 · 𝐵) · (𝐸 · 𝐴)) = ((𝐷 · 𝐴) · (𝐸 · 𝐵))) | |
24 | 10, 12, 18, 8, 23 | syl22anc 836 | . . . . 5 ⊢ (𝜑 → ((𝐷 · 𝐵) · (𝐸 · 𝐴)) = ((𝐷 · 𝐴) · (𝐸 · 𝐵))) |
25 | 24 | oveq2d 7165 | . . . 4 ⊢ (𝜑 → (2 · ((𝐷 · 𝐵) · (𝐸 · 𝐴))) = (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) |
26 | 22, 25 | oveq12d 7167 | . . 3 ⊢ (𝜑 → (((𝐷 · 𝐵)↑2) + (2 · ((𝐷 · 𝐵) · (𝐸 · 𝐴)))) = (((𝐷↑2) · (𝐵↑2)) + (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵))))) |
27 | 18, 8 | sqmuld 13519 | . . 3 ⊢ (𝜑 → ((𝐸 · 𝐴)↑2) = ((𝐸↑2) · (𝐴↑2))) |
28 | 26, 27 | oveq12d 7167 | . 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 2859 | 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 1536 ∈ wcel 2113 (class class class)co 7149 ℂcc 10528 ℝcr 10529 + caddc 10533 · cmul 10535 − cmin 10863 2c2 11686 ↑cexp 13426 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-nn 11632 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13367 df-exp 13427 |
This theorem is referenced by: 2itscplem3 44841 |
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