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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 10245. (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 8828 | . . . . . 6 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℕ0) |
4 | 3 | nn0cnd 8826 | . . . . 5 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℂ) |
5 | 4 | sqvald 10198 | . . . 4 ⊢ (𝜑 → ((𝐴 + 𝐵)↑2) = ((𝐴 + 𝐵) · (𝐴 + 𝐵))) |
6 | 5 | oveq1d 5705 | . . 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 8828 | . . . . . 6 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℕ0) |
10 | 9 | nn0cnd 8826 | . . . . 5 ⊢ (𝜑 → (𝐶 + 𝐷) ∈ ℂ) |
11 | 10 | sqvald 10198 | . . . 4 ⊢ (𝜑 → ((𝐶 + 𝐷)↑2) = ((𝐶 + 𝐷) · (𝐶 + 𝐷))) |
12 | 11 | oveq1d 5705 | . . 3 ⊢ (𝜑 → (((𝐶 + 𝐷)↑2) + 𝐷) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷)) |
13 | 6, 12 | eqeq12d 2109 | . 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 10245 | . 2 ⊢ (𝜑 → ((((𝐴 + 𝐵) · (𝐴 + 𝐵)) + 𝐵) = (((𝐶 + 𝐷) · (𝐶 + 𝐷)) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))) |
15 | 13, 14 | bitrd 187 | 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 103 ↔ wb 104 = wceq 1296 ∈ wcel 1445 (class class class)co 5690 + caddc 7450 · cmul 7452 2c2 8571 ℕ0cn0 8771 ↑cexp 10069 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-if 3414 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-tr 3959 df-id 4144 df-po 4147 df-iso 4148 df-iord 4217 df-on 4219 df-ilim 4220 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-recs 6108 df-frec 6194 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 df-inn 8521 df-2 8579 df-n0 8772 df-z 8849 df-uz 9119 df-seqfrec 10001 df-exp 10070 |
This theorem is referenced by: nn0opth2 10247 |
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