Theorem 1kp2ke3k 14829

Description: Example for df-dec 9399, 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 9399 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 9206	. . . 4 |- 1 e. NN0
2		0nn0 9205	. . . 4 |- 0 e. NN0
3	1, 2	deccl 9412	. . 3 |- ; 1 0 e. NN0
4	3, 2	deccl 9412	. 2 |- ;; 1 0 0 e. NN0
5		2nn0 9207	. . . 4 |- 2 e. NN0
6	5, 2	deccl 9412	. . 3 |- ; 2 0 e. NN0
7	6, 2	deccl 9412	. 2 |- ;; 2 0 0 e. NN0
8		eqid 2187	. 2 |- ;;; 1 0 0 0 = ;;; 1 0 0 0
9		eqid 2187	. 2 |- ;;; 2 0 0 0 = ;;; 2 0 0 0
10		eqid 2187	. . 3 |- ;; 1 0 0 = ;; 1 0 0
11		eqid 2187	. . 3 |- ;; 2 0 0 = ;; 2 0 0
12		eqid 2187	. . . 4 |- ; 1 0 = ; 1 0
13		eqid 2187	. . . 4 |- ; 2 0 = ; 2 0
14		1p2e3 9067	. . . 4 |- ( 1 + 2 ) = 3
15		00id 8112	. . . 4 |- ( 0 + 0 ) = 0
16	1, 2, 5, 2, 12, 13, 14, 15	decadd 9451	. . 3 |- (; 1 0 + ; 2 0 ) = ; 3 0
17	3, 2, 6, 2, 10, 11, 16, 15	decadd 9451	. 2 |- (;; 1 0 0 + ;; 2 0 0 ) = ;; 3 0 0
18	4, 2, 7, 2, 8, 9, 17, 15	decadd 9451	1 |- (;;; 1 0 0 0 + ;;; 2 0 0 0 ) = ;;; 3 0 0 0

Syntax hints:   = wceq 1363  (class class class)co 5888   0cc0 7825   1c1 7826   + caddc 7828   2c2 8984   3c3 8985  ;cdc 9398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-cnre 7936

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-sub 8144  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-5 8995  df-6 8996  df-7 8997  df-8 8998  df-9 8999  df-n0 9191  df-dec 9399

This theorem is referenced by: (None)