Theorem 1kp2ke3k 30348

Description: Example for df-dec 12626, 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 12626 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 12434	. . . 4 ⊢ 1 ∈ ℕ0
2		0nn0 12433	. . . 4 ⊢ 0 ∈ ℕ0
3	1, 2	deccl 12640	. . 3 ⊢ ;10 ∈ ℕ0
4	3, 2	deccl 12640	. 2 ⊢ ;;100 ∈ ℕ0
5		2nn0 12435	. . . 4 ⊢ 2 ∈ ℕ0
6	5, 2	deccl 12640	. . 3 ⊢ ;20 ∈ ℕ0
7	6, 2	deccl 12640	. 2 ⊢ ;;200 ∈ ℕ0
8		eqid 2729	. 2 ⊢ ;;;1000 = ;;;1000
9		eqid 2729	. 2 ⊢ ;;;2000 = ;;;2000
10		eqid 2729	. . 3 ⊢ ;;100 = ;;100
11		eqid 2729	. . 3 ⊢ ;;200 = ;;200
12		eqid 2729	. . . 4 ⊢ ;10 = ;10
13		eqid 2729	. . . 4 ⊢ ;20 = ;20
14		1p2e3 12300	. . . 4 ⊢ (1 + 2) = 3
15		00id 11325	. . . 4 ⊢ (0 + 0) = 0
16	1, 2, 5, 2, 12, 13, 14, 15	decadd 12679	. . 3 ⊢ (;10 + ;20) = ;30
17	3, 2, 6, 2, 10, 11, 16, 15	decadd 12679	. 2 ⊢ (;;100 + ;;200) = ;;300
18	4, 2, 7, 2, 8, 9, 17, 15	decadd 12679	1 ⊢ (;;;1000 + ;;;2000) = ;;;3000

Syntax hints:   = wceq 1540  (class class class)co 7369  0cc0 11044  1c1 11045   + caddc 11047  2c2 12217  3c3 12218  ;cdc 12625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-resscn 11101  ax-1cn 11102  ax-icn 11103  ax-addcl 11104  ax-addrcl 11105  ax-mulcl 11106  ax-mulrcl 11107  ax-mulcom 11108  ax-addass 11109  ax-mulass 11110  ax-distr 11111  ax-i2m1 11112  ax-1ne0 11113  ax-1rid 11114  ax-rnegex 11115  ax-rrecex 11116  ax-cnre 11117  ax-pre-lttri 11118  ax-pre-lttrn 11119  ax-pre-ltadd 11120

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-ltxr 11189  df-nn 12163  df-2 12225  df-3 12226  df-4 12227  df-5 12228  df-6 12229  df-7 12230  df-8 12231  df-9 12232  df-n0 12419  df-dec 12626

This theorem is referenced by: (None)