Theorem nn0opth2 14228

Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 14226. (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 7411	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (𝐴 + 𝐵) = (if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵))
2	1	oveq1d 7419	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → ((𝐴 + 𝐵)↑2) = ((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2))
3	2	oveq1d 7419	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (((𝐴 + 𝐵)↑2) + 𝐵) = (((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) + 𝐵))
4	3	eqeq1d 2735	. . 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 2737	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (𝐴 = 𝐶 ↔ if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶))
6	5	anbi1d 631	. . 3 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ 𝐵 = 𝐷)))
7	4, 6	bibi12d 346	. 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 7412	. . . . . 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 7419	. . . . 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 22	. . . . 5 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → 𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0))
11	9, 10	oveq12d 7422	. . . 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 2735	. . 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 2737	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → (𝐵 = 𝐷 ↔ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷))
14	13	anbi2d 630	. . 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 346	. 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 7411	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (𝐶 + 𝐷) = (if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷))
17	16	oveq1d 7419	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → ((𝐶 + 𝐷)↑2) = ((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2))
18	17	oveq1d 7419	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (((𝐶 + 𝐷)↑2) + 𝐷) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) + 𝐷))
19	18	eqeq2d 2744	. . 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 2745	. . . 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 631	. . 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 346	. 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 7412	. . . . . 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 7419	. . . . 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 22	. . . . 5 ⊢ (𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0) → 𝐷 = if(𝐷 ∈ ℕ0, 𝐷, 0))
26	24, 25	oveq12d 7422	. . . 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 2744	. . 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 2745	. . . 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 630	. . 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 346	. 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 12483	. . . 4 ⊢ 0 ∈ ℕ0
32	31	elimel 4596	. . 3 ⊢ if(𝐴 ∈ ℕ0, 𝐴, 0) ∈ ℕ0
33	31	elimel 4596	. . 3 ⊢ if(𝐵 ∈ ℕ0, 𝐵, 0) ∈ ℕ0
34	31	elimel 4596	. . 3 ⊢ if(𝐶 ∈ ℕ0, 𝐶, 0) ∈ ℕ0
35	31	elimel 4596	. . 3 ⊢ if(𝐷 ∈ ℕ0, 𝐷, 0) ∈ ℕ0
36	32, 33, 34, 35	nn0opth2i 14227	. 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 4588	1 ⊢ (((𝐴 ∈ ℕ0 ∧ 𝐵 ∈ ℕ0) ∧ (𝐶 ∈ ℕ0 ∧ 𝐷 ∈ ℕ0)) → ((((𝐴 + 𝐵)↑2) + 𝐵) = (((𝐶 + 𝐷)↑2) + 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  ifcif 4527  (class class class)co 7404  0cc0 11106   + caddc 11109  2c2 12263  ℕ₀cn0 12468  ↑cexp 14023

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-seq 13963  df-exp 14024

This theorem is referenced by: (None)