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Mirrors > Home > MPE Home > Th. List > nn0opth2i | Structured version Visualization version GIF version |
Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthi 14124. (Contributed by NM, 22-Jul-2004.) |
Ref | Expression |
---|---|
nn0opth.1 | ⊢ 𝐴 ∈ ℕ0 |
nn0opth.2 | ⊢ 𝐵 ∈ ℕ0 |
nn0opth.3 | ⊢ 𝐶 ∈ ℕ0 |
nn0opth.4 | ⊢ 𝐷 ∈ ℕ0 |
Ref | Expression |
---|---|
nn0opth2i | ⊢ ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0opth.1 | . . . . . . 7 ⊢ 𝐴 ∈ ℕ0 | |
2 | 1 | nn0cni 12383 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
3 | nn0opth.2 | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
4 | 3 | nn0cni 12383 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
5 | 2, 4 | addcli 11119 | . . . . 5 ⊢ (𝐴 + 𝐵) ∈ ℂ |
6 | 5 | sqvali 14038 | . . . 4 ⊢ ((𝐴 + 𝐵)↑2) = ((𝐴 + 𝐵) · (𝐴 + 𝐵)) |
7 | 6 | oveq1i 7361 | . . 3 ⊢ (((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) |
8 | nn0opth.3 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 | |
9 | 8 | nn0cni 12383 | . . . . . 6 ⊢ 𝐶 ∈ ℂ |
10 | nn0opth.4 | . . . . . . 7 ⊢ 𝐷 ∈ ℕ0 | |
11 | 10 | nn0cni 12383 | . . . . . 6 ⊢ 𝐷 ∈ ℂ |
12 | 9, 11 | addcli 11119 | . . . . 5 ⊢ (𝐶 + 𝐷) ∈ ℂ |
13 | 12 | sqvali 14038 | . . . 4 ⊢ ((𝐶 + 𝐷)↑2) = ((𝐶 + 𝐷) · (𝐶 + 𝐷)) |
14 | 13 | oveq1i 7361 | . . 3 ⊢ (((𝐶 + 𝐷)↑2) + 𝐷) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) |
15 | 7, 14 | eqeq12i 2755 | . 2 ⊢ ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷)) |
16 | 1, 3, 8, 10 | nn0opthi 14124 | . 2 ⊢ ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
17 | 15, 16 | bitri 274 | 1 ⊢ ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 (class class class)co 7351 + caddc 11012 · cmul 11014 2c2 12166 ℕ0cn0 12371 ↑cexp 13921 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-n0 12372 df-z 12458 df-uz 12722 df-seq 13861 df-exp 13922 |
This theorem is referenced by: nn0opth2 14126 |
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