Theorem nn0opth2 14172

Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opthi 14170. (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 7364	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (𝐴 + 𝐵) = (if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵))
2	1	oveq1d 7372	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → ((𝐴 + 𝐵)↑2) = ((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2))
3	2	oveq1d 7372	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (((𝐴 + 𝐵)↑2) + 𝐵) = (((if(𝐴 ∈ ℕ0, 𝐴, 0) + 𝐵)↑2) + 𝐵))
4	3	eqeq1d 2738	. . 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 2740	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ ℕ0, 𝐴, 0) → (𝐴 = 𝐶 ↔ if(𝐴 ∈ ℕ0, 𝐴, 0) = 𝐶))
6	5	anbi1d 630	. . 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 7365	. . . . . 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 7372	. . . . 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 7375	. . . 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 2738	. . 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 2740	. . . 4 ⊢ (𝐵 = if(𝐵 ∈ ℕ0, 𝐵, 0) → (𝐵 = 𝐷 ↔ if(𝐵 ∈ ℕ0, 𝐵, 0) = 𝐷))
14	13	anbi2d 629	. . 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 7364	. . . . . 6 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (𝐶 + 𝐷) = (if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷))
17	16	oveq1d 7372	. . . . 5 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → ((𝐶 + 𝐷)↑2) = ((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2))
18	17	oveq1d 7372	. . . 4 ⊢ (𝐶 = if(𝐶 ∈ ℕ0, 𝐶, 0) → (((𝐶 + 𝐷)↑2) + 𝐷) = (((if(𝐶 ∈ ℕ0, 𝐶, 0) + 𝐷)↑2) + 𝐷))
19	18	eqeq2d 2747	. . 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 2748	. . . 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 630	. . 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 7365	. . . . . 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 7372	. . . . 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 7375	. . . 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 2747	. . 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 2748	. . . 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 629	. . 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 12428	. . . 4 ⊢ 0 ∈ ℕ0
32	31	elimel 4555	. . 3 ⊢ if(𝐴 ∈ ℕ0, 𝐴, 0) ∈ ℕ0
33	31	elimel 4555	. . 3 ⊢ if(𝐵 ∈ ℕ0, 𝐵, 0) ∈ ℕ0
34	31	elimel 4555	. . 3 ⊢ if(𝐶 ∈ ℕ0, 𝐶, 0) ∈ ℕ0
35	31	elimel 4555	. . 3 ⊢ if(𝐷 ∈ ℕ0, 𝐷, 0) ∈ ℕ0
36	32, 33, 34, 35	nn0opth2i 14171	. 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 4547	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 396   = wceq 1541   ∈ wcel 2106  ifcif 4486  (class class class)co 7357  0cc0 11051   + caddc 11054  2c2 12208  ℕ₀cn0 12413  ↑cexp 13967

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-n0 12414  df-z 12500  df-uz 12764  df-seq 13907  df-exp 13968

This theorem is referenced by: (None)