Theorem 1kp2ke3k 13342

Description: Example for df-dec 9297, 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 9297 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	|- (;;; 1 0 0 0 + ;;; 2 0 0 0 ) = ;;; 3 0 0 0

Proof of Theorem 1kp2ke3k

Step	Hyp	Ref	Expression
1		1nn0 9107	. . . 4 |- 1 e. NN0
2		0nn0 9106	. . . 4 |- 0 e. NN0
3	1, 2	deccl 9310	. . 3 |- ; 1 0 e. NN0
4	3, 2	deccl 9310	. 2 |- ;; 1 0 0 e. NN0
5		2nn0 9108	. . . 4 |- 2 e. NN0
6	5, 2	deccl 9310	. . 3 |- ; 2 0 e. NN0
7	6, 2	deccl 9310	. 2 |- ;; 2 0 0 e. NN0
8		eqid 2157	. 2 |- ;;; 1 0 0 0 = ;;; 1 0 0 0
9		eqid 2157	. 2 |- ;;; 2 0 0 0 = ;;; 2 0 0 0
10		eqid 2157	. . 3 |- ;; 1 0 0 = ;; 1 0 0
11		eqid 2157	. . 3 |- ;; 2 0 0 = ;; 2 0 0
12		eqid 2157	. . . 4 |- ; 1 0 = ; 1 0
13		eqid 2157	. . . 4 |- ; 2 0 = ; 2 0
14		1p2e3 8968	. . . 4 |- ( 1 + 2 ) = 3
15		00id 8017	. . . 4 |- ( 0 + 0 ) = 0
16	1, 2, 5, 2, 12, 13, 14, 15	decadd 9349	. . 3 |- (; 1 0 + ; 2 0 ) = ; 3 0
17	3, 2, 6, 2, 10, 11, 16, 15	decadd 9349	. 2 |- (;; 1 0 0 + ;; 2 0 0 ) = ;; 3 0 0
18	4, 2, 7, 2, 8, 9, 17, 15	decadd 9349	1 |- (;;; 1 0 0 0 + ;;; 2 0 0 0 ) = ;;; 3 0 0 0

Syntax hints:   = wceq 1335  (class class class)co 5825   0cc0 7733   1c1 7734   + caddc 7736   2c2 8885   3c3 8886  ;cdc 9296

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4083  ax-pow 4136  ax-pr 4170  ax-setind 4497  ax-cnex 7824  ax-resscn 7825  ax-1cn 7826  ax-1re 7827  ax-icn 7828  ax-addcl 7829  ax-addrcl 7830  ax-mulcl 7831  ax-addcom 7833  ax-mulcom 7834  ax-addass 7835  ax-mulass 7836  ax-distr 7837  ax-i2m1 7838  ax-1rid 7840  ax-0id 7841  ax-rnegex 7842  ax-cnre 7844

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3774  df-int 3809  df-br 3967  df-opab 4027  df-id 4254  df-xp 4593  df-rel 4594  df-cnv 4595  df-co 4596  df-dm 4597  df-iota 5136  df-fun 5173  df-fv 5179  df-riota 5781  df-ov 5828  df-oprab 5829  df-mpo 5830  df-sub 8049  df-inn 8835  df-2 8893  df-3 8894  df-4 8895  df-5 8896  df-6 8897  df-7 8898  df-8 8899  df-9 8900  df-n0 9092  df-dec 9297

This theorem is referenced by: (None)