Theorem nn0opth2 14416

Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 14414. (Contributed by NM, 22-Jul-2004.)

Assertion

Ref	Expression
nn0opth2	⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0)) → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Proof of Theorem nn0opth2

Step	Hyp	Ref	Expression
1		oveq1 7427	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (𝐴 + 𝐵) = (if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵))
2	1	oveq1d 7435	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → ((𝐴 + 𝐵)↑2) = ((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2))
3	2	oveq1d 7435	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (((𝐴 + 𝐵)↑2) + 𝐵) = (((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) + 𝐵))
4	3	eqeq1d 2763	. . 3 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷)))
5		eqeq1 2765	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (𝐴 = 𝐶 ↔ if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶))
6	5	anbi1d 643	. . 3 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ 𝐵 = 𝐷)))
7	4, 6	bibi12d 348	. 2 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) ↔ ((((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ 𝐵 = 𝐷))))
8		oveq2 7428	. . . . . 6 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → (if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵) = (if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0)))
9	8	oveq1d 7435	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → ((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) = ((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2))
10		id 23	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → 𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0))
11	9, 10	oveq12d 7438	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → (((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) + 𝐵) = (((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)))
12	11	eqeq1d 2763	. . 3 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → ((((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((𝐶 + 𝐷)↑2) + 𝐷)))
13		eqeq1 2765	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → (𝐵 = 𝐷 ↔ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷))
14	13	anbi2d 642	. . 3 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → ((if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ 𝐵 = 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷)))
15	12, 14	bibi12d 348	. 2 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → (((((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ 𝐵 = 𝐷)) ↔ ((((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷))))
16		oveq1 7427	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (𝐶 + 𝐷) = (if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷))
17	16	oveq1d 7435	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → ((𝐶 + 𝐷)↑2) = ((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2))
18	17	oveq1d 7435	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (((𝐶 + 𝐷)↑2) + 𝐷) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) + 𝐷))
19	18	eqeq2d 2772	. . 3 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → ((((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) + 𝐷)))
20		eqeq2 2773	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ↔ if(𝐴 ∈ ℕ0, 𝐴, 0) = if(𝐶 ∈ ℕ0, 𝐶, 0)))
21	20	anbi1d 643	. . 3 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → ((if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = if(𝐶 ∈ ℕ0, 𝐶, 0) ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷)))
22	19, 21	bibi12d 348	. 2 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (((((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷)) ↔ ((((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) + 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = if(𝐶 ∈ ℕ0, 𝐶, 0) ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷))))
23		oveq2 7428	. . . . . 6 ⊢ (𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0) → (if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷) = (if(𝐶 ∈ ℕ0, 𝐶, 0) + if(𝐷 ∈ ℕ0, 𝐷, 0)))
24	23	oveq1d 7435	. . . . 5 ⊢ (𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0) → ((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) = ((if(𝐶 ∈ ℕ0, 𝐶, 0) + if(𝐷 ∈ ℕ0, 𝐷, 0))↑2))
25		id 23	. . . . 5 ⊢ (𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0) → 𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0))
26	24, 25	oveq12d 7438	. . . 4 ⊢ (𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0) → (((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) + 𝐷) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + if(𝐷 ∈ ℕ0, 𝐷, 0))↑2) + if(𝐷 ∈ ℕ0, 𝐷, 0)))
27	26	eqeq2d 2772	. . 3 ⊢ (𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0) → ((((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) + 𝐷) ↔ (((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + if(𝐷 ∈ ℕ0, 𝐷, 0))↑2) + if(𝐷 ∈ ℕ0, 𝐷, 0))))
28		eqeq2 2773	. . . 4 ⊢ (𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0) → (if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷 ↔ if(𝐵 ∈ ℕ0, 𝐵, 0) = if(𝐷 ∈ ℕ0, 𝐷, 0)))
29	28	anbi2d 642	. . 3 ⊢ (𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0) → ((if(𝐴 ∈ ℕ0, 𝐴, 0) = if(𝐶 ∈ ℕ0, 𝐶, 0) ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = if(𝐶 ∈ ℕ0, 𝐶, 0) ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = if(𝐷 ∈ ℕ0, 𝐷, 0))))
30	27, 29	bibi12d 348	. 2 ⊢ (𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0) → (((((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) + 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = if(𝐶 ∈ ℕ0, 𝐶, 0) ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷)) ↔ ((((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + if(𝐷 ∈ ℕ0, 𝐷, 0))↑2) + if(𝐷 ∈ ℕ0, 𝐷, 0)) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = if(𝐶 ∈ ℕ0, 𝐶, 0) ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = if(𝐷 ∈ ℕ0, 𝐷, 0)))))
31		0nn0 12621	. . . 4 ⊢ 0 ∈ ℕ0
32	31	elimel 4552	. . 3 ⊢ if(𝐴 ∈ ℕ0, 𝐴, 0) ∈ ℕ0
33	31	elimel 4552	. . 3 ⊢ if(𝐵 ∈ ℕ0, 𝐵, 0) ∈ ℕ0
34	31	elimel 4552	. . 3 ⊢ if(𝐶 ∈ ℕ0, 𝐶, 0) ∈ ℕ0
35	31	elimel 4552	. . 3 ⊢ if(𝐷 ∈ ℕ0, 𝐷, 0) ∈ ℕ0
36	32, 33, 34, 35	nn0opth2i 14415	. 2 ⊢ ((((if(𝐴 ∈ ℕ0, 𝐴, 0) + if(𝐵 ∈ ℕ0, 𝐵, 0))↑2) + if(𝐵 ∈ ℕ0, 𝐵, 0)) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + if(𝐷 ∈ ℕ0, 𝐷, 0))↑2) + if(𝐷 ∈ ℕ0, 𝐷, 0)) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = if(𝐶 ∈ ℕ0, 𝐶, 0) ∧ if(𝐵 ∈ ℕ0, 𝐵, 0) = if(𝐷 ∈ ℕ0, 𝐷, 0)))
37	7, 15, 22, 30, 36	dedth4h 4544	1 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0)) → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ifcif 4482  (class class class)co 7420  0cc0 11200   + caddc 11203  2c2 12397  ℕ₀cn0 12606  ↑cexp 14204

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751  ax-cnex 11256  ax-resscn 11257  ax-1cn 11258  ax-icn 11259  ax-addcl 11260  ax-addrcl 11261  ax-mulcl 11262  ax-mulrcl 11263  ax-mulcom 11264  ax-addass 11265  ax-mulass 11266  ax-distr 11267  ax-i2m1 11268  ax-1ne0 11269  ax-1rid 11270  ax-rnegex 11271  ax-rrecex 11272  ax-cnre 11273  ax-pre-lttri 11274  ax-pre-lttrn 11275  ax-pre-ltadd 11276  ax-pre-mulgt0 11277

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-pnf 11345  df-mnf 11346  df-xr 11347  df-ltxr 11348  df-le 11349  df-sub 11543  df-neg 11544  df-nn 12336  df-2 12405  df-n0 12607  df-z 12694  df-uz 12966  df-seq 14145  df-exp 14205

This theorem is used by: (None)