Theorem nn0opth2 14225

Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 14223. (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 7367	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (𝐴 + 𝐵) = (if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵))
2	1	oveq1d 7375	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → ((𝐴 + 𝐵)↑2) = ((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2))
3	2	oveq1d 7375	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (((𝐴 + 𝐵)↑2) + 𝐵) = (((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) + 𝐵))
4	3	eqeq1d 2739	. . 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 2741	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (𝐴 = 𝐶 ↔ if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶))
6	5	anbi1d 632	. . 3 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ (if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶 ∧ 𝐵 = 𝐷)))
7	4, 6	bibi12d 345	. 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 7368	. . . . . 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 7375	. . . . 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 7378	. . . 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 2739	. . 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 2741	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → (𝐵 = 𝐷 ↔ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷))
14	13	anbi2d 631	. . 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 345	. 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 7367	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (𝐶 + 𝐷) = (if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷))
17	16	oveq1d 7375	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → ((𝐶 + 𝐷)↑2) = ((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2))
18	17	oveq1d 7375	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (((𝐶 + 𝐷)↑2) + 𝐷) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) + 𝐷))
19	18	eqeq2d 2748	. . 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 2749	. . . 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 632	. . 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 345	. 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 7368	. . . . . 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 7375	. . . . 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 7378	. . . 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 2748	. . 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 2749	. . . 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 631	. . 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 345	. 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 12443	. . . 4 ⊢ 0 ∈ ℕ0
32	31	elimel 4537	. . 3 ⊢ if(𝐴 ∈ ℕ0, 𝐴, 0) ∈ ℕ0
33	31	elimel 4537	. . 3 ⊢ if(𝐵 ∈ ℕ0, 𝐵, 0) ∈ ℕ0
34	31	elimel 4537	. . 3 ⊢ if(𝐶 ∈ ℕ0, 𝐶, 0) ∈ ℕ0
35	31	elimel 4537	. . 3 ⊢ if(𝐷 ∈ ℕ0, 𝐷, 0) ∈ ℕ0
36	32, 33, 34, 35	nn0opth2i 14224	. 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 4529	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ifcif 4467  (class class class)co 7360  0cc0 11029   + caddc 11032  2c2 12227  ℕ₀cn0 12428  ↑cexp 14014

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682  ax-cnex 11085  ax-resscn 11086  ax-1cn 11087  ax-icn 11088  ax-addcl 11089  ax-addrcl 11090  ax-mulcl 11091  ax-mulrcl 11092  ax-mulcom 11093  ax-addass 11094  ax-mulass 11095  ax-distr 11096  ax-i2m1 11097  ax-1ne0 11098  ax-1rid 11099  ax-rnegex 11100  ax-rrecex 11101  ax-cnre 11102  ax-pre-lttri 11103  ax-pre-lttrn 11104  ax-pre-ltadd 11105  ax-pre-mulgt0 11106

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6259  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-rdg 8342  df-er 8636  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-sub 11370  df-neg 11371  df-nn 12166  df-2 12235  df-n0 12429  df-z 12516  df-uz 12780  df-seq 13955  df-exp 14015

This theorem is referenced by: (None)