Theorem nn0opth2 14271

Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 14269. (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 7433	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (𝐴 + 𝐵) = (if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵))
2	1	oveq1d 7441	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → ((𝐴 + 𝐵)↑2) = ((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2))
3	2	oveq1d 7441	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (((𝐴 + 𝐵)↑2) + 𝐵) = (((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) + 𝐵))
4	3	eqeq1d 2730	. . 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 2732	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (𝐴 = 𝐶 ↔ if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶))
6	5	anbi1d 629	. . 3 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ 𝐵 = 𝐷)))
7	4, 6	bibi12d 344	. 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 7434	. . . . . 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 7441	. . . . 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 7444	. . . 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 2730	. . 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 2732	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → (𝐵 = 𝐷 ↔ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷))
14	13	anbi2d 628	. . 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 344	. 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 7433	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (𝐶 + 𝐷) = (if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷))
17	16	oveq1d 7441	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → ((𝐶 + 𝐷)↑2) = ((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2))
18	17	oveq1d 7441	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (((𝐶 + 𝐷)↑2) + 𝐷) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) + 𝐷))
19	18	eqeq2d 2739	. . 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 2740	. . . 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 629	. . 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 344	. 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 7434	. . . . . 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 7441	. . . . 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 7444	. . . 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 2739	. . 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 2740	. . . 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 628	. . 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 344	. 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 12525	. . . 4 ⊢ 0 ∈ ℕ0
32	31	elimel 4601	. . 3 ⊢ if(𝐴 ∈ ℕ0, 𝐴, 0) ∈ ℕ0
33	31	elimel 4601	. . 3 ⊢ if(𝐵 ∈ ℕ0, 𝐵, 0) ∈ ℕ0
34	31	elimel 4601	. . 3 ⊢ if(𝐶 ∈ ℕ0, 𝐶, 0) ∈ ℕ0
35	31	elimel 4601	. . 3 ⊢ if(𝐷 ∈ ℕ0, 𝐷, 0) ∈ ℕ0
36	32, 33, 34, 35	nn0opth2i 14270	. 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 4593	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 394   = wceq 1533   ∈ wcel 2098  ifcif 4532  (class class class)co 7426  0cc0 11146   + caddc 11149  2c2 12305  ℕ₀cn0 12510  ↑cexp 14066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-tr 5270  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6310  df-ord 6377  df-on 6378  df-lim 6379  df-suc 6380  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-om 7877  df-2nd 8000  df-frecs 8293  df-wrecs 8324  df-recs 8398  df-rdg 8437  df-er 8731  df-en 8971  df-dom 8972  df-sdom 8973  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-nn 12251  df-2 12313  df-n0 12511  df-z 12597  df-uz 12861  df-seq 14007  df-exp 14067

This theorem is referenced by: (None)

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