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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 12919. (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 11181 | . . . . . 6 ⊢ 𝐴 ∈ ℂ |
3 | nn0opth.2 | . . . . . . 7 ⊢ 𝐵 ∈ ℕ0 | |
4 | 3 | nn0cni 11181 | . . . . . 6 ⊢ 𝐵 ∈ ℂ |
5 | 2, 4 | addcli 9923 | . . . . 5 ⊢ (𝐴 + 𝐵) ∈ ℂ |
6 | 5 | sqvali 12805 | . . . 4 ⊢ ((𝐴 + 𝐵)↑2) = ((𝐴 + 𝐵) · (𝐴 + 𝐵)) |
7 | 6 | oveq1i 6559 | . . 3 ⊢ (((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) |
8 | nn0opth.3 | . . . . . . 7 ⊢ 𝐶 ∈ ℕ0 | |
9 | 8 | nn0cni 11181 | . . . . . 6 ⊢ 𝐶 ∈ ℂ |
10 | nn0opth.4 | . . . . . . 7 ⊢ 𝐷 ∈ ℕ0 | |
11 | 10 | nn0cni 11181 | . . . . . 6 ⊢ 𝐷 ∈ ℂ |
12 | 9, 11 | addcli 9923 | . . . . 5 ⊢ (𝐶 + 𝐷) ∈ ℂ |
13 | 12 | sqvali 12805 | . . . 4 ⊢ ((𝐶 + 𝐷)↑2) = ((𝐶 + 𝐷) · (𝐶 + 𝐷)) |
14 | 13 | oveq1i 6559 | . . 3 ⊢ (((𝐶 + 𝐷)↑2) + 𝐷) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) |
15 | 7, 14 | eqeq12i 2624 | . 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 12919 | . 2 ⊢ ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
17 | 15, 16 | bitri 263 | 1 ⊢ ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 (class class class)co 6549 + caddc 9818 · cmul 9820 2c2 10947 ℕ0cn0 11169 ↑cexp 12722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-seq 12664 df-exp 12723 |
This theorem is referenced by: nn0opth2 12921 |
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