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Mirrors > Home > ILE Home > Th. List > nn0opth2d | GIF version |
Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See comments for nn0opthd 9979. (Contributed by Jim Kingdon, 31-Oct-2021.) |
Ref | Expression |
---|---|
nn0opthd.1 | ⊢ (𝜑 → 𝐴 ∈ ℕ0) |
nn0opthd.2 | ⊢ (𝜑 → 𝐵 ∈ ℕ0) |
nn0opthd.3 | ⊢ (𝜑 → 𝐶 ∈ ℕ0) |
nn0opthd.4 | ⊢ (𝜑 → 𝐷 ∈ ℕ0) |
Ref | Expression |
---|---|
nn0opth2d | ⊢ (𝜑 → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nn0opthd.1 | . . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℕ0) | |
2 | nn0opthd.2 | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℕ0) | |
3 | 1, 2 | nn0addcld 8640 | . . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ0) |
4 | 3 | nn0cnd 8638 | . . . . 5 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
5 | 4 | sqvald 9932 | . . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵)↑2) = ((𝐴 + 𝐵) · (𝐴 + 𝐵))) |
6 | 5 | oveq1d 5609 | . . 3 ⊢ (𝜑 → (((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵)) |
7 | nn0opthd.3 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ ℕ0) | |
8 | nn0opthd.4 | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ ℕ0) | |
9 | 7, 8 | nn0addcld 8640 | . . . . . 6 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℕ0) |
10 | 9 | nn0cnd 8638 | . . . . 5 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℂ) |
11 | 10 | sqvald 9932 | . . . 4 ⊢ (𝜑 → ((𝐶 + 𝐷)↑2) = ((𝐶 + 𝐷) · (𝐶 + 𝐷))) |
12 | 11 | oveq1d 5609 | . . 3 ⊢ (𝜑 → (((𝐶 + 𝐷)↑2) + 𝐷) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷)) |
13 | 6, 12 | eqeq12d 2099 | . 2 ⊢ (𝜑 → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷))) |
14 | 1, 2, 7, 8 | nn0opthd 9979 | . 2 ⊢ (𝜑 → ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
15 | 13, 14 | bitrd 186 | 1 ⊢ (𝜑 → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1287 ∈ wcel 1436 (class class class)co 5594 + caddc 7274 · cmul 7276 2c2 8384 ℕ0cn0 8583 ↑cexp 9805 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-13 1447 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-coll 3922 ax-sep 3925 ax-nul 3933 ax-pow 3977 ax-pr 4003 ax-un 4227 ax-setind 4319 ax-iinf 4369 ax-cnex 7357 ax-resscn 7358 ax-1cn 7359 ax-1re 7360 ax-icn 7361 ax-addcl 7362 ax-addrcl 7363 ax-mulcl 7364 ax-mulrcl 7365 ax-addcom 7366 ax-mulcom 7367 ax-addass 7368 ax-mulass 7369 ax-distr 7370 ax-i2m1 7371 ax-0lt1 7372 ax-1rid 7373 ax-0id 7374 ax-rnegex 7375 ax-precex 7376 ax-cnre 7377 ax-pre-ltirr 7378 ax-pre-ltwlin 7379 ax-pre-lttrn 7380 ax-pre-apti 7381 ax-pre-ltadd 7382 ax-pre-mulgt0 7383 ax-pre-mulext 7384 |
This theorem depends on definitions: df-bi 115 df-dc 779 df-3or 923 df-3an 924 df-tru 1290 df-fal 1293 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ne 2252 df-nel 2347 df-ral 2360 df-rex 2361 df-reu 2362 df-rmo 2363 df-rab 2364 df-v 2616 df-sbc 2829 df-csb 2922 df-dif 2988 df-un 2990 df-in 2992 df-ss 2999 df-nul 3273 df-if 3377 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-uni 3631 df-int 3666 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-tr 3905 df-id 4087 df-po 4090 df-iso 4091 df-iord 4160 df-on 4162 df-ilim 4163 df-suc 4165 df-iom 4372 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-res 4416 df-ima 4417 df-iota 4937 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 df-fv 4980 df-riota 5550 df-ov 5597 df-oprab 5598 df-mpt2 5599 df-1st 5849 df-2nd 5850 df-recs 6005 df-frec 6091 df-pnf 7445 df-mnf 7446 df-xr 7447 df-ltxr 7448 df-le 7449 df-sub 7576 df-neg 7577 df-reap 7970 df-ap 7977 df-div 8056 df-inn 8335 df-2 8393 df-n0 8584 df-z 8661 df-uz 8929 df-iseq 9755 df-iexp 9806 |
This theorem is referenced by: nn0opth2 9981 |
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