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Mirrors > Home > MPE Home > Th. List > Mathboxes > sqmid3api | Structured version Visualization version GIF version |
Description: Value of the square of the middle term of a 3-term arithmetic progression. (Contributed by Steven Nguyen, 20-Sep-2022.) |
Ref | Expression |
---|---|
sqmid3api.a | ⊢ 𝐴 ∈ ℂ |
sqmid3api.n | ⊢ 𝑁 ∈ ℂ |
sqmid3api.b | ⊢ (𝐴 + 𝑁) = 𝐵 |
sqmid3api.c | ⊢ (𝐵 + 𝑁) = 𝐶 |
Ref | Expression |
---|---|
sqmid3api | ⊢ (𝐵 · 𝐵) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sqmid3api.a | . . 3 ⊢ 𝐴 ∈ ℂ | |
2 | sqmid3api.n | . . 3 ⊢ 𝑁 ∈ ℂ | |
3 | 1, 2, 1, 2 | muladdi 11080 | . 2 ⊢ ((𝐴 + 𝑁) · (𝐴 + 𝑁)) = (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
4 | sqmid3api.b | . . 3 ⊢ (𝐴 + 𝑁) = 𝐵 | |
5 | 4, 4 | oveq12i 7147 | . 2 ⊢ ((𝐴 + 𝑁) · (𝐴 + 𝑁)) = (𝐵 · 𝐵) |
6 | 1, 1 | mulcli 10637 | . . . 4 ⊢ (𝐴 · 𝐴) ∈ ℂ |
7 | 2, 2 | mulcli 10637 | . . . 4 ⊢ (𝑁 · 𝑁) ∈ ℂ |
8 | 1, 2 | mulcli 10637 | . . . . 5 ⊢ (𝐴 · 𝑁) ∈ ℂ |
9 | 8, 8 | addcli 10636 | . . . 4 ⊢ ((𝐴 · 𝑁) + (𝐴 · 𝑁)) ∈ ℂ |
10 | 6, 7, 9 | add32i 10852 | . . 3 ⊢ (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = (((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) + (𝑁 · 𝑁)) |
11 | 1, 2 | addcli 10636 | . . . . . 6 ⊢ (𝐴 + 𝑁) ∈ ℂ |
12 | 1, 11, 2 | adddii 10642 | . . . . 5 ⊢ (𝐴 · ((𝐴 + 𝑁) + 𝑁)) = ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) |
13 | 4 | oveq1i 7145 | . . . . . . 7 ⊢ ((𝐴 + 𝑁) + 𝑁) = (𝐵 + 𝑁) |
14 | sqmid3api.c | . . . . . . 7 ⊢ (𝐵 + 𝑁) = 𝐶 | |
15 | 13, 14 | eqtri 2821 | . . . . . 6 ⊢ ((𝐴 + 𝑁) + 𝑁) = 𝐶 |
16 | 15 | oveq2i 7146 | . . . . 5 ⊢ (𝐴 · ((𝐴 + 𝑁) + 𝑁)) = (𝐴 · 𝐶) |
17 | 1, 1, 2 | adddii 10642 | . . . . . . 7 ⊢ (𝐴 · (𝐴 + 𝑁)) = ((𝐴 · 𝐴) + (𝐴 · 𝑁)) |
18 | 17 | oveq1i 7145 | . . . . . 6 ⊢ ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) = (((𝐴 · 𝐴) + (𝐴 · 𝑁)) + (𝐴 · 𝑁)) |
19 | 6, 8, 8 | addassi 10640 | . . . . . 6 ⊢ (((𝐴 · 𝐴) + (𝐴 · 𝑁)) + (𝐴 · 𝑁)) = ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
20 | 18, 19 | eqtri 2821 | . . . . 5 ⊢ ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) = ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
21 | 12, 16, 20 | 3eqtr3ri 2830 | . . . 4 ⊢ ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = (𝐴 · 𝐶) |
22 | 21 | oveq1i 7145 | . . 3 ⊢ (((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) + (𝑁 · 𝑁)) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
23 | 10, 22 | eqtri 2821 | . 2 ⊢ (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
24 | 3, 5, 23 | 3eqtr3i 2829 | 1 ⊢ (𝐵 · 𝐵) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 (class class class)co 7135 ℂcc 10524 + caddc 10529 · cmul 10531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 |
This theorem is referenced by: sqn5i 39479 |
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