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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 11079 | . 2 ⊢ ((𝐴 + 𝑁) · (𝐴 + 𝑁)) = (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
4 | sqmid3api.b | . . 3 ⊢ (𝐴 + 𝑁) = 𝐵 | |
5 | 4, 4 | oveq12i 7157 | . 2 ⊢ ((𝐴 + 𝑁) · (𝐴 + 𝑁)) = (𝐵 · 𝐵) |
6 | 1, 1 | mulcli 10636 | . . . 4 ⊢ (𝐴 · 𝐴) ∈ ℂ |
7 | 2, 2 | mulcli 10636 | . . . 4 ⊢ (𝑁 · 𝑁) ∈ ℂ |
8 | 1, 2 | mulcli 10636 | . . . . 5 ⊢ (𝐴 · 𝑁) ∈ ℂ |
9 | 8, 8 | addcli 10635 | . . . 4 ⊢ ((𝐴 · 𝑁) + (𝐴 · 𝑁)) ∈ ℂ |
10 | 6, 7, 9 | add32i 10851 | . . 3 ⊢ (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = (((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) + (𝑁 · 𝑁)) |
11 | 1, 2 | addcli 10635 | . . . . . 6 ⊢ (𝐴 + 𝑁) ∈ ℂ |
12 | 1, 11, 2 | adddii 10641 | . . . . 5 ⊢ (𝐴 · ((𝐴 + 𝑁) + 𝑁)) = ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) |
13 | 4 | oveq1i 7155 | . . . . . . 7 ⊢ ((𝐴 + 𝑁) + 𝑁) = (𝐵 + 𝑁) |
14 | sqmid3api.c | . . . . . . 7 ⊢ (𝐵 + 𝑁) = 𝐶 | |
15 | 13, 14 | eqtri 2841 | . . . . . 6 ⊢ ((𝐴 + 𝑁) + 𝑁) = 𝐶 |
16 | 15 | oveq2i 7156 | . . . . 5 ⊢ (𝐴 · ((𝐴 + 𝑁) + 𝑁)) = (𝐴 · 𝐶) |
17 | 1, 1, 2 | adddii 10641 | . . . . . . 7 ⊢ (𝐴 · (𝐴 + 𝑁)) = ((𝐴 · 𝐴) + (𝐴 · 𝑁)) |
18 | 17 | oveq1i 7155 | . . . . . 6 ⊢ ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) = (((𝐴 · 𝐴) + (𝐴 · 𝑁)) + (𝐴 · 𝑁)) |
19 | 6, 8, 8 | addassi 10639 | . . . . . 6 ⊢ (((𝐴 · 𝐴) + (𝐴 · 𝑁)) + (𝐴 · 𝑁)) = ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
20 | 18, 19 | eqtri 2841 | . . . . 5 ⊢ ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) = ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
21 | 12, 16, 20 | 3eqtr3ri 2850 | . . . 4 ⊢ ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = (𝐴 · 𝐶) |
22 | 21 | oveq1i 7155 | . . 3 ⊢ (((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) + (𝑁 · 𝑁)) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
23 | 10, 22 | eqtri 2841 | . 2 ⊢ (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
24 | 3, 5, 23 | 3eqtr3i 2849 | 1 ⊢ (𝐵 · 𝐵) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 (class class class)co 7145 ℂcc 10523 + caddc 10528 · cmul 10530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 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-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-ltxr 10668 |
This theorem is referenced by: sqn5i 39049 |
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