Theorem 1kp2ke3k 13759

Description: Example for df-dec 9344, 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 9344 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 9151	. . . 4 ⊢ 1 ∈ ℕ0
2		0nn0 9150	. . . 4 ⊢ 0 ∈ ℕ0
3	1, 2	deccl 9357	. . 3 ⊢ ;10 ∈ ℕ0
4	3, 2	deccl 9357	. 2 ⊢ ;;100 ∈ ℕ0
5		2nn0 9152	. . . 4 ⊢ 2 ∈ ℕ0
6	5, 2	deccl 9357	. . 3 ⊢ ;20 ∈ ℕ0
7	6, 2	deccl 9357	. 2 ⊢ ;;200 ∈ ℕ0
8		eqid 2170	. 2 ⊢ ;;;1000 = ;;;1000
9		eqid 2170	. 2 ⊢ ;;;2000 = ;;;2000
10		eqid 2170	. . 3 ⊢ ;;100 = ;;100
11		eqid 2170	. . 3 ⊢ ;;200 = ;;200
12		eqid 2170	. . . 4 ⊢ ;10 = ;10
13		eqid 2170	. . . 4 ⊢ ;20 = ;20
14		1p2e3 9012	. . . 4 ⊢ (1 + 2) = 3
15		00id 8060	. . . 4 ⊢ (0 + 0) = 0
16	1, 2, 5, 2, 12, 13, 14, 15	decadd 9396	. . 3 ⊢ (;10 + ;20) = ;30
17	3, 2, 6, 2, 10, 11, 16, 15	decadd 9396	. 2 ⊢ (;;100 + ;;200) = ;;300
18	4, 2, 7, 2, 8, 9, 17, 15	decadd 9396	1 ⊢ (;;;1000 + ;;;2000) = ;;;3000

Syntax hints:   = wceq 1348  (class class class)co 5853  0cc0 7774  1c1 7775   + caddc 7777  2c2 8929  3c3 8930  ;cdc 9343

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-sub 8092  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-9 8944  df-n0 9136  df-dec 9344

This theorem is referenced by: (None)