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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 11084 | . 2 ⊢ ((𝐴 + 𝑁) · (𝐴 + 𝑁)) = (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
4 | sqmid3api.b | . . 3 ⊢ (𝐴 + 𝑁) = 𝐵 | |
5 | 4, 4 | oveq12i 7161 | . 2 ⊢ ((𝐴 + 𝑁) · (𝐴 + 𝑁)) = (𝐵 · 𝐵) |
6 | 1, 1 | mulcli 10641 | . . . 4 ⊢ (𝐴 · 𝐴) ∈ ℂ |
7 | 2, 2 | mulcli 10641 | . . . 4 ⊢ (𝑁 · 𝑁) ∈ ℂ |
8 | 1, 2 | mulcli 10641 | . . . . 5 ⊢ (𝐴 · 𝑁) ∈ ℂ |
9 | 8, 8 | addcli 10640 | . . . 4 ⊢ ((𝐴 · 𝑁) + (𝐴 · 𝑁)) ∈ ℂ |
10 | 6, 7, 9 | add32i 10856 | . . 3 ⊢ (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = (((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) + (𝑁 · 𝑁)) |
11 | 1, 2 | addcli 10640 | . . . . . 6 ⊢ (𝐴 + 𝑁) ∈ ℂ |
12 | 1, 11, 2 | adddii 10646 | . . . . 5 ⊢ (𝐴 · ((𝐴 + 𝑁) + 𝑁)) = ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) |
13 | 4 | oveq1i 7159 | . . . . . . 7 ⊢ ((𝐴 + 𝑁) + 𝑁) = (𝐵 + 𝑁) |
14 | sqmid3api.c | . . . . . . 7 ⊢ (𝐵 + 𝑁) = 𝐶 | |
15 | 13, 14 | eqtri 2843 | . . . . . 6 ⊢ ((𝐴 + 𝑁) + 𝑁) = 𝐶 |
16 | 15 | oveq2i 7160 | . . . . 5 ⊢ (𝐴 · ((𝐴 + 𝑁) + 𝑁)) = (𝐴 · 𝐶) |
17 | 1, 1, 2 | adddii 10646 | . . . . . . 7 ⊢ (𝐴 · (𝐴 + 𝑁)) = ((𝐴 · 𝐴) + (𝐴 · 𝑁)) |
18 | 17 | oveq1i 7159 | . . . . . 6 ⊢ ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) = (((𝐴 · 𝐴) + (𝐴 · 𝑁)) + (𝐴 · 𝑁)) |
19 | 6, 8, 8 | addassi 10644 | . . . . . 6 ⊢ (((𝐴 · 𝐴) + (𝐴 · 𝑁)) + (𝐴 · 𝑁)) = ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
20 | 18, 19 | eqtri 2843 | . . . . 5 ⊢ ((𝐴 · (𝐴 + 𝑁)) + (𝐴 · 𝑁)) = ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) |
21 | 12, 16, 20 | 3eqtr3ri 2852 | . . . 4 ⊢ ((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = (𝐴 · 𝐶) |
22 | 21 | oveq1i 7159 | . . 3 ⊢ (((𝐴 · 𝐴) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) + (𝑁 · 𝑁)) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
23 | 10, 22 | eqtri 2843 | . 2 ⊢ (((𝐴 · 𝐴) + (𝑁 · 𝑁)) + ((𝐴 · 𝑁) + (𝐴 · 𝑁))) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
24 | 3, 5, 23 | 3eqtr3i 2851 | 1 ⊢ (𝐵 · 𝐵) = ((𝐴 · 𝐶) + (𝑁 · 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2113 (class class class)co 7149 ℂcc 10528 + caddc 10533 · cmul 10535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-mpt 5140 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 7152 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-pnf 10670 df-mnf 10671 df-ltxr 10673 |
This theorem is referenced by: sqn5i 39247 |
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