Theorem nn0opth2t 6669

Description: An ordered pair theorem for nonnegative integers. Theorem 17.3 of [Quine] p. 124. See nn0opth 6667.

Assertion

Ref	Expression
nn0opth2t	⊢ (((A ∈ ℕ0 ⋀ B ∈ ℕ0) ⋀ (C ∈ ℕ0 ⋀ D ∈ ℕ0)) → ((((A + B)↑2) + B) = (((C + D)↑2) + D) ↔ (A = C ⋀ B = D)))

Proof of Theorem nn0opth2t

Step	Hyp	Ref	Expression
1		opreq1 3974	. . . . . 6 ⊢ (A = if(A ∈ ℕ0, A, 0) → (A + B) = ( if(A ∈ ℕ0, A, 0) + B))
2	1	opreq1d 3981	. . . . 5 ⊢ (A = if(A ∈ ℕ0, A, 0) → ((A + B)↑2) = (( if(A ∈ ℕ0, A, 0) + B)↑2))
3	2	opreq1d 3981	. . . 4 ⊢ (A = if(A ∈ ℕ0, A, 0) → (((A + B)↑2) + B) = ((( if(A ∈ ℕ0, A, 0) + B)↑2) + B))
4	3	eqeq1d 1486	. . 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 1484	. . . 4 ⊢ (A = if(A ∈ ℕ0, A, 0) → (A = C ↔ if(A ∈ ℕ0, A, 0) = C))
6	5	anbi1d 619	. . 3 ⊢ (A = if(A ∈ ℕ0, A, 0) → ((A = C ⋀ B = D) ↔ ( if(A ∈ ℕ0, A, 0) = C ⋀ B = D)))
7	4, 6	bibi12d 631	. 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		opreq2 3975	. . . . . 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	opreq1d 3981	. . . . 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 59	. . . . 5 ⊢ (B = if(B ∈ ℕ0, B, 0) → B = if(B ∈ ℕ0, B, 0))
11	9, 10	opreq12d 3984	. . . 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 1486	. . 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 1484	. . . 4 ⊢ (B = if(B ∈ ℕ0, B, 0) → (B = D ↔ if(B ∈ ℕ0, B, 0) = D))
14	13	anbi2d 618	. . 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 631	. 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		opreq1 3974	. . . . . 6 ⊢ (C = if(C ∈ ℕ0, C, 0) → (C + D) = ( if(C ∈ ℕ0, C, 0) + D))
17	16	opreq1d 3981	. . . . 5 ⊢ (C = if(C ∈ ℕ0, C, 0) → ((C + D)↑2) = (( if(C ∈ ℕ0, C, 0) + D)↑2))
18	17	opreq1d 3981	. . . 4 ⊢ (C = if(C ∈ ℕ0, C, 0) → (((C + D)↑2) + D) = ((( if(C ∈ ℕ0, C, 0) + D)↑2) + D))
19	18	eqeq2d 1489	. . 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 1487	. . . 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 619	. . 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 631	. 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		opreq2 3975	. . . . . 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	opreq1d 3981	. . . . 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 59	. . . . 5 ⊢ (D = if(D ∈ ℕ0, D, 0) → D = if(D ∈ ℕ0, D, 0))
26	24, 25	opreq12d 3984	. . . 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 1489	. . 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 1487	. . . 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 618	. . 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 631	. 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 6115	. . . 4 ⊢ 0 ∈ ℕ0
32	31	elimel 2398	. . 3 ⊢ if(A ∈ ℕ0, A, 0) ∈ ℕ0
33	31	elimel 2398	. . 3 ⊢ if(B ∈ ℕ0, B, 0) ∈ ℕ0
34	31	elimel 2398	. . 3 ⊢ if(C ∈ ℕ0, C, 0) ∈ ℕ0
35	31	elimel 2398	. . 3 ⊢ if(D ∈ ℕ0, D, 0) ∈ ℕ0
36	32, 33, 34, 35	nn0opth2 6668	. 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 2393	1 ⊢ (((A ∈ ℕ0 ⋀ B ∈ ℕ0) ⋀ (C ∈ ℕ0 ⋀ D ∈ ℕ0)) → ((((A + B)↑2) + B) = (((C + D)↑2) + D) ↔ (A = C ⋀ B = D)))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960   ifcif 2365  (class class class)co 3969  0cc0 5246   + caddc 5249  ℕ₀cn0 5309  2c2 5963  ↑cexp 6569

This theorem is referenced by:  xpnnen 7500

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-mp 5101  df-ltp 5102  df-plpr 5176  df-mpr 5177  df-enr 5178  df-nr 5179  df-plr 5180  df-mr 5181  df-ltr 5182  df-0r 5183  df-1r 5184  df-m1r 5185  df-c 5252  df-0 5253  df-1 5254  df-i 5255  df-r 5256  df-plus 5257  df-mul 5258  df-lt 5259  df-sub 5368  df-neg 5370  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-le 5503  df-n 5927  df-2 5972  df-n0 6102  df-z 6138  df-seq1 6309  df-exp 6570

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