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Mirrors > Home > MPE Home > Th. List > 1kp2ke3k | Structured version Visualization version GIF version |
Description: Example for df-dec 11850, 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 11850 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 11664 | . . . 4 ⊢ 1 ∈ ℕ0 | |
2 | 0nn0 11663 | . . . 4 ⊢ 0 ∈ ℕ0 | |
3 | 1, 2 | deccl 11864 | . . 3 ⊢ ;10 ∈ ℕ0 |
4 | 3, 2 | deccl 11864 | . 2 ⊢ ;;100 ∈ ℕ0 |
5 | 2nn0 11665 | . . . 4 ⊢ 2 ∈ ℕ0 | |
6 | 5, 2 | deccl 11864 | . . 3 ⊢ ;20 ∈ ℕ0 |
7 | 6, 2 | deccl 11864 | . 2 ⊢ ;;200 ∈ ℕ0 |
8 | eqid 2778 | . 2 ⊢ ;;;1000 = ;;;1000 | |
9 | eqid 2778 | . 2 ⊢ ;;;2000 = ;;;2000 | |
10 | eqid 2778 | . . 3 ⊢ ;;100 = ;;100 | |
11 | eqid 2778 | . . 3 ⊢ ;;200 = ;;200 | |
12 | eqid 2778 | . . . 4 ⊢ ;10 = ;10 | |
13 | eqid 2778 | . . . 4 ⊢ ;20 = ;20 | |
14 | 1p2e3 11529 | . . . 4 ⊢ (1 + 2) = 3 | |
15 | 00id 10553 | . . . 4 ⊢ (0 + 0) = 0 | |
16 | 1, 2, 5, 2, 12, 13, 14, 15 | decadd 11904 | . . 3 ⊢ (;10 + ;20) = ;30 |
17 | 3, 2, 6, 2, 10, 11, 16, 15 | decadd 11904 | . 2 ⊢ (;;100 + ;;200) = ;;300 |
18 | 4, 2, 7, 2, 8, 9, 17, 15 | decadd 11904 | 1 ⊢ (;;;1000 + ;;;2000) = ;;;3000 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1601 (class class class)co 6924 0cc0 10274 1c1 10275 + caddc 10277 2c2 11434 3c3 11435 ;cdc 11849 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-ov 6927 df-om 7346 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-er 8028 df-en 8244 df-dom 8245 df-sdom 8246 df-pnf 10415 df-mnf 10416 df-ltxr 10418 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-dec 11850 |
This theorem is referenced by: (None) |
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