![]() |
Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
|
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 10772 | . 2 ⊢ ((𝐴 + 𝑁) · (𝐴 + 𝑁)) = (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
4 | sqmid3api.b | . . 3 ⊢ (𝐴 + 𝑁) = 𝐵 | |
5 | 4, 4 | oveq12i 6889 | . 2 ⊢ ((𝐴 + 𝑁) · (𝐴 + 𝑁)) = (𝐵 · 𝐵) |
6 | 1, 1 | mulcli 10335 | . . . 4 ⊢ (𝐴 · 𝐴) ∈ ℂ |
7 | 2, 2 | mulcli 10335 | . . . 4 ⊢ (𝑁 · 𝑁) ∈ ℂ |
8 | 1, 2 | mulcli 10335 | . . . . 5 ⊢ (𝐴 · 𝑁) ∈ ℂ |
9 | 8, 8 | addcli 10334 | . . . 4 ⊢ ((𝐴 · 𝑁) + (𝐴 · 𝑁)) ∈ ℂ |
10 | 6, 7, 9 | add32i 10548 | . . 3 ⊢ (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = (((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) + (𝑁 · 𝑁)) |
11 | 1, 2 | addcli 10334 | . . . . . 6 ⊢ (𝐴 + 𝑁) ∈ ℂ |
12 | 1, 11, 2 | adddii 10340 | . . . . 5 ⊢ (𝐴 · ((𝐴 + 𝑁) + 𝑁)) = ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) |
13 | 4 | oveq1i 6887 | . . . . . . 7 ⊢ ((𝐴 + 𝑁) + 𝑁) = (𝐵 + 𝑁) |
14 | sqmid3api.c | . . . . . . 7 ⊢ (𝐵 + 𝑁) = 𝐶 | |
15 | 13, 14 | eqtri 2820 | . . . . . 6 ⊢ ((𝐴 + 𝑁) + 𝑁) = 𝐶 |
16 | 15 | oveq2i 6888 | . . . . 5 ⊢ (𝐴 · ((𝐴 + 𝑁) + 𝑁)) = (𝐴 · 𝐶) |
17 | 1, 1, 2 | adddii 10340 | . . . . . . 7 ⊢ (𝐴 · (𝐴 + 𝑁)) = ((𝐴 · 𝐴) + (𝐴 · 𝑁)) |
18 | 17 | oveq1i 6887 | . . . . . 6 ⊢ ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) = (((𝐴 · 𝐴) + (𝐴 · 𝑁)) + (𝐴 · 𝑁)) |
19 | 6, 8, 8 | addassi 10338 | . . . . . 6 ⊢ (((𝐴 · 𝐴) + (𝐴 · 𝑁)) + (𝐴 · 𝑁)) = ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
20 | 18, 19 | eqtri 2820 | . . . . 5 ⊢ ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) = ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
21 | 12, 16, 20 | 3eqtr3ri 2829 | . . . 4 ⊢ ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = (𝐴 · 𝐶) |
22 | 21 | oveq1i 6887 | . . 3 ⊢ (((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) + (𝑁 · 𝑁)) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
23 | 10, 22 | eqtri 2820 | . 2 ⊢ (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
24 | 3, 5, 23 | 3eqtr3i 2828 | 1 ⊢ (𝐵 · 𝐵) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1653 ∈ wcel 2157 (class class class)co 6877 ℂcc 10221 + caddc 10226 · cmul 10228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2776 ax-sep 4974 ax-nul 4982 ax-pow 5034 ax-pr 5096 ax-un 7182 ax-resscn 10280 ax-1cn 10281 ax-icn 10282 ax-addcl 10283 ax-addrcl 10284 ax-mulcl 10285 ax-mulrcl 10286 ax-mulcom 10287 ax-addass 10288 ax-mulass 10289 ax-distr 10290 ax-i2m1 10291 ax-1ne0 10292 ax-1rid 10293 ax-rnegex 10294 ax-rrecex 10295 ax-cnre 10296 ax-pre-lttri 10297 ax-pre-lttrn 10298 ax-pre-ltadd 10299 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2785 df-cleq 2791 df-clel 2794 df-nfc 2929 df-ne 2971 df-nel 3074 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3386 df-sbc 3633 df-csb 3728 df-dif 3771 df-un 3773 df-in 3775 df-ss 3782 df-nul 4115 df-if 4277 df-pw 4350 df-sn 4368 df-pr 4370 df-op 4374 df-uni 4628 df-br 4843 df-opab 4905 df-mpt 4922 df-id 5219 df-po 5232 df-so 5233 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-iota 6063 df-fun 6102 df-fn 6103 df-f 6104 df-f1 6105 df-fo 6106 df-f1o 6107 df-fv 6108 df-ov 6880 df-er 7981 df-en 8195 df-dom 8196 df-sdom 8197 df-pnf 10364 df-mnf 10365 df-ltxr 10367 |
This theorem is referenced by: sqn5i 37987 |
Copyright terms: Public domain | W3C validator |