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Mirrors > Home > MPE Home > Th. List > 1kp2ke3k | Structured version Visualization version GIF version |
Description: Example for df-dec 12087, 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 12087 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.) |
Ref | Expression |
---|---|
1kp2ke3k | ⊢ (;;;1000 + ;;;2000) = ;;;3000 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1nn0 11901 | . . . 4 ⊢ 1 ∈ ℕ0 | |
2 | 0nn0 11900 | . . . 4 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 12101 | . . 3 ⊢ ;10 ∈ ℕ0 |
4 | 3, 2 | deccl 12101 | . 2 ⊢ ;;100 ∈ ℕ0 |
5 | 2nn0 11902 | . . . 4 ⊢ 2 ∈ ℕ0 | |
6 | 5, 2 | deccl 12101 | . . 3 ⊢ ;20 ∈ ℕ0 |
7 | 6, 2 | deccl 12101 | . 2 ⊢ ;;200 ∈ ℕ0 |
8 | eqid 2798 | . 2 ⊢ ;;;1000 = ;;;1000 | |
9 | eqid 2798 | . 2 ⊢ ;;;2000 = ;;;2000 | |
10 | eqid 2798 | . . 3 ⊢ ;;100 = ;;100 | |
11 | eqid 2798 | . . 3 ⊢ ;;200 = ;;200 | |
12 | eqid 2798 | . . . 4 ⊢ ;10 = ;10 | |
13 | eqid 2798 | . . . 4 ⊢ ;20 = ;20 | |
14 | 1p2e3 11768 | . . . 4 ⊢ (1 + 2) = 3 | |
15 | 00id 10804 | . . . 4 ⊢ (0 + 0) = 0 | |
16 | 1, 2, 5, 2, 12, 13, 14, 15 | decadd 12140 | . . 3 ⊢ (;10 + ;20) = ;30 |
17 | 3, 2, 6, 2, 10, 11, 16, 15 | decadd 12140 | . 2 ⊢ (;;100 + ;;200) = ;;300 |
18 | 4, 2, 7, 2, 8, 9, 17, 15 | decadd 12140 | 1 ⊢ (;;;1000 + ;;;2000) = ;;;3000 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 (class class class)co 7135 0cc0 10526 1c1 10527 + caddc 10529 2c2 11680 3c3 11681 ;cdc 12086 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-om 7561 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-ltxr 10669 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-dec 12087 |
This theorem is referenced by: (None) |
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