Theorem 1kp2ke3k 30424

Description: Example for df-dec 12589, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with (2 + 1) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 12589 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

Assertion

Ref	Expression
1kp2ke3k	⊢ (;;;1000 + ;;;2000) = ;;;3000

Proof of Theorem 1kp2ke3k

Step	Hyp	Ref	Expression
1		1nn0 12397	. . . 4 ⊢ 1 ∈ ℕ0
2		0nn0 12396	. . . 4 ⊢ 0 ∈ ℕ0
3	1, 2	deccl 12603	. . 3 ⊢ ;10 ∈ ℕ0
4	3, 2	deccl 12603	. 2 ⊢ ;;100 ∈ ℕ0
5		2nn0 12398	. . . 4 ⊢ 2 ∈ ℕ0
6	5, 2	deccl 12603	. . 3 ⊢ ;20 ∈ ℕ0
7	6, 2	deccl 12603	. 2 ⊢ ;;200 ∈ ℕ0
8		eqid 2731	. 2 ⊢ ;;;1000 = ;;;1000
9		eqid 2731	. 2 ⊢ ;;;2000 = ;;;2000
10		eqid 2731	. . 3 ⊢ ;;100 = ;;100
11		eqid 2731	. . 3 ⊢ ;;200 = ;;200
12		eqid 2731	. . . 4 ⊢ ;10 = ;10
13		eqid 2731	. . . 4 ⊢ ;20 = ;20
14		1p2e3 12263	. . . 4 ⊢ (1 + 2) = 3
15		00id 11288	. . . 4 ⊢ (0 + 0) = 0
16	1, 2, 5, 2, 12, 13, 14, 15	decadd 12642	. . 3 ⊢ (;10 + ;20) = ;30
17	3, 2, 6, 2, 10, 11, 16, 15	decadd 12642	. 2 ⊢ (;;100 + ;;200) = ;;300
18	4, 2, 7, 2, 8, 9, 17, 15	decadd 12642	1 ⊢ (;;;1000 + ;;;2000) = ;;;3000

Syntax hints:   = wceq 1541  (class class class)co 7346  0cc0 11006  1c1 11007   + caddc 11009  2c2 12180  3c3 12181  ;cdc 12588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-om 7797  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-ltxr 11151  df-nn 12126  df-2 12188  df-3 12189  df-4 12190  df-5 12191  df-6 12192  df-7 12193  df-8 12194  df-9 12195  df-n0 12382  df-dec 12589

This theorem is referenced by: (None)