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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 11697 | . 2 ⊢ ((𝐴 + 𝑁) · (𝐴 + 𝑁)) = (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
4 | sqmid3api.b | . . 3 ⊢ (𝐴 + 𝑁) = 𝐵 | |
5 | 4, 4 | oveq12i 7431 | . 2 ⊢ ((𝐴 + 𝑁) · (𝐴 + 𝑁)) = (𝐵 · 𝐵) |
6 | 1, 1 | mulcli 11253 | . . . 4 ⊢ (𝐴 · 𝐴) ∈ ℂ |
7 | 2, 2 | mulcli 11253 | . . . 4 ⊢ (𝑁 · 𝑁) ∈ ℂ |
8 | 1, 2 | mulcli 11253 | . . . . 5 ⊢ (𝐴 · 𝑁) ∈ ℂ |
9 | 8, 8 | addcli 11252 | . . . 4 ⊢ ((𝐴 · 𝑁) + (𝐴 · 𝑁)) ∈ ℂ |
10 | 6, 7, 9 | add32i 11469 | . . 3 ⊢ (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = (((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) + (𝑁 · 𝑁)) |
11 | 1, 2 | addcli 11252 | . . . . . 6 ⊢ (𝐴 + 𝑁) ∈ ℂ |
12 | 1, 11, 2 | adddii 11258 | . . . . 5 ⊢ (𝐴 · ((𝐴 + 𝑁) + 𝑁)) = ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) |
13 | 4 | oveq1i 7429 | . . . . . . 7 ⊢ ((𝐴 + 𝑁) + 𝑁) = (𝐵 + 𝑁) |
14 | sqmid3api.c | . . . . . . 7 ⊢ (𝐵 + 𝑁) = 𝐶 | |
15 | 13, 14 | eqtri 2753 | . . . . . 6 ⊢ ((𝐴 + 𝑁) + 𝑁) = 𝐶 |
16 | 15 | oveq2i 7430 | . . . . 5 ⊢ (𝐴 · ((𝐴 + 𝑁) + 𝑁)) = (𝐴 · 𝐶) |
17 | 1, 1, 2 | adddii 11258 | . . . . . . 7 ⊢ (𝐴 · (𝐴 + 𝑁)) = ((𝐴 · 𝐴) + (𝐴 · 𝑁)) |
18 | 17 | oveq1i 7429 | . . . . . 6 ⊢ ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) = (((𝐴 · 𝐴) + (𝐴 · 𝑁)) + (𝐴 · 𝑁)) |
19 | 6, 8, 8 | addassi 11256 | . . . . . 6 ⊢ (((𝐴 · 𝐴) + (𝐴 · 𝑁)) + (𝐴 · 𝑁)) = ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
20 | 18, 19 | eqtri 2753 | . . . . 5 ⊢ ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) = ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
21 | 12, 16, 20 | 3eqtr3ri 2762 | . . . 4 ⊢ ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = (𝐴 · 𝐶) |
22 | 21 | oveq1i 7429 | . . 3 ⊢ (((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) + (𝑁 · 𝑁)) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
23 | 10, 22 | eqtri 2753 | . 2 ⊢ (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
24 | 3, 5, 23 | 3eqtr3i 2761 | 1 ⊢ (𝐵 · 𝐵) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 (class class class)co 7419 ℂcc 11138 + caddc 11143 · cmul 11145 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-ltxr 11285 |
This theorem is referenced by: sqn5i 41994 |
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