Theorem 1kp2ke3k 15622

Description: Example for df-dec 9504, 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 9504 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 9310	. . . 4 |- 1 e. NN0
2		0nn0 9309	. . . 4 |- 0 e. NN0
3	1, 2	deccl 9517	. . 3 |- ; 1 0 e. NN0
4	3, 2	deccl 9517	. 2 |- ;; 1 0 0 e. NN0
5		2nn0 9311	. . . 4 |- 2 e. NN0
6	5, 2	deccl 9517	. . 3 |- ; 2 0 e. NN0
7	6, 2	deccl 9517	. 2 |- ;; 2 0 0 e. NN0
8		eqid 2204	. 2 |- ;;; 1 0 0 0 = ;;; 1 0 0 0
9		eqid 2204	. 2 |- ;;; 2 0 0 0 = ;;; 2 0 0 0
10		eqid 2204	. . 3 |- ;; 1 0 0 = ;; 1 0 0
11		eqid 2204	. . 3 |- ;; 2 0 0 = ;; 2 0 0
12		eqid 2204	. . . 4 |- ; 1 0 = ; 1 0
13		eqid 2204	. . . 4 |- ; 2 0 = ; 2 0
14		1p2e3 9170	. . . 4 |- ( 1 + 2 ) = 3
15		00id 8212	. . . 4 |- ( 0 + 0 ) = 0
16	1, 2, 5, 2, 12, 13, 14, 15	decadd 9556	. . 3 |- (; 1 0 + ; 2 0 ) = ; 3 0
17	3, 2, 6, 2, 10, 11, 16, 15	decadd 9556	. 2 |- (;; 1 0 0 + ;; 2 0 0 ) = ;; 3 0 0
18	4, 2, 7, 2, 8, 9, 17, 15	decadd 9556	1 |- (;;; 1 0 0 0 + ;;; 2 0 0 0 ) = ;;; 3 0 0 0

Syntax hints:   = wceq 1372  (class class class)co 5943   0cc0 7924   1c1 7925   + caddc 7927   2c2 9086   3c3 9087  ;cdc 9503

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-iota 5231  df-fun 5272  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-sub 8244  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-5 9097  df-6 9098  df-7 9099  df-8 9100  df-9 9101  df-n0 9295  df-dec 9504

This theorem is referenced by: (None)