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Mirrors > Home > MPE Home > Th. List > Mathboxes > 2itscplem2 | Structured version Visualization version GIF version |
Description: Lemma 2 for 2itscp 44788. (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 7166 | . . 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 10669 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℂ) |
7 | 2itscp.a | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
8 | 7 | recnd 10669 | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
9 | 6, 8 | subcld 10997 | . . . . 5 ⊢ (𝜑 → (𝑋 − 𝐴) ∈ ℂ) |
10 | 4, 9 | eqeltrid 2917 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
11 | 2itscp.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
12 | 11 | recnd 10669 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
13 | 10, 12 | mulcld 10661 | . . 3 ⊢ (𝜑 → (𝐷 · 𝐵) ∈ ℂ) |
14 | 2itscp.e | . . . . 5 ⊢ 𝐸 = (𝐵 − 𝑌) | |
15 | 2itscp.y | . . . . . . 7 ⊢ (𝜑 → 𝑌 ∈ ℝ) | |
16 | 15 | recnd 10669 | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ℂ) |
17 | 12, 16 | subcld 10997 | . . . . 5 ⊢ (𝜑 → (𝐵 − 𝑌) ∈ ℂ) |
18 | 14, 17 | eqeltrid 2917 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℂ) |
19 | 18, 8 | mulcld 10661 | . . 3 ⊢ (𝜑 → (𝐸 · 𝐴) ∈ ℂ) |
20 | binom2 13580 | . . 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 13523 | . . . 4 ⊢ (𝜑 → ((𝐷 · 𝐵)↑2) = ((𝐷↑2) · (𝐵↑2))) |
23 | mul4r 10809 | . . . . . 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 7172 | . . . 4 ⊢ (𝜑 → (2 · ((𝐷 · 𝐵) · (𝐸 · 𝐴))) = (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵)))) |
26 | 22, 25 | oveq12d 7174 | . . 3 ⊢ (𝜑 → (((𝐷 · 𝐵)↑2) + (2 · ((𝐷 · 𝐵) · (𝐸 · 𝐴)))) = (((𝐷↑2) · (𝐵↑2)) + (2 · ((𝐷 · 𝐴) · (𝐸 · 𝐵))))) |
27 | 18, 8 | sqmuld 13523 | . . 3 ⊢ (𝜑 → ((𝐸 · 𝐴)↑2) = ((𝐸↑2) · (𝐴↑2))) |
28 | 26, 27 | oveq12d 7174 | . 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 2860 | 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 1537 ∈ wcel 2114 (class class class)co 7156 ℂcc 10535 ℝcr 10536 + caddc 10540 · cmul 10542 − cmin 10870 2c2 11693 ↑cexp 13430 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-n0 11899 df-z 11983 df-uz 12245 df-seq 13371 df-exp 13431 |
This theorem is referenced by: 2itscplem3 44787 |
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