Theorem dec10 8397

Description: The decimal form of 10. NB: In our presentations of large numbers later on, we will use our symbol for 10 at the highest digits when advantageous, because we can use this theorem to convert back to "long form" (where each digit is in the range 0-9) with no extra effort. However, we cannot do this for lower digits while maintaining the ease of use of the decimal system, since it requires nontrivial number knowledge (more than just equality theorems) to convert back. (Contributed by Mario Carneiro, 18-Feb-2014.)

Assertion

Ref	Expression
dec10	⊢ 10 = ;10

Proof of Theorem dec10

Step	Hyp	Ref	Expression
1		10nn 8085	. . . 4 ⊢ 10 ∈ ℕ
2	1	nncni 7924	. . 3 ⊢ 10 ∈ ℂ
3	2	addid1i 7155	. 2 ⊢ (10 + 0) = 10
4		dec10p 8396	. 2 ⊢ (10 + 0) = ;10
5	3, 4	eqtr3i 2062	1 ⊢ 10 = ;10

Syntax hints:   = wceq 1243  (class class class)co 5512  0cc0 6889  1c1 6890   + caddc 6892  10c10 7972  ;cdc 8368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-mulcom 6985  ax-mulass 6987  ax-distr 6988  ax-1rid 6991  ax-0id 6992  ax-cnre 6995

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-iota 4867  df-fv 4910  df-ov 5515  df-inn 7915  df-2 7973  df-3 7974  df-4 7975  df-5 7976  df-6 7977  df-7 7978  df-8 7979  df-9 7980  df-10 7981  df-dec 8369

This theorem is referenced by:  decaddc2  8410  decaddci2  8413  6p5e11  8417  7p4e11  8419  8p3e11  8423  9p2e11  8429  10p10e20  8437