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Mirrors > Home > ILE Home > Th. List > 9p1e10 | GIF version |
Description: 9 + 1 = 10. (Contributed by Mario Carneiro, 18-Apr-2015.) (Revised by Stanislas Polu, 7-Apr-2020.) (Revised by AV, 1-Aug-2021.) |
Ref | Expression |
---|---|
9p1e10 | ⊢ (9 + 1) = ;10 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dec 9314 | . 2 ⊢ ;10 = (((9 + 1) · 1) + 0) | |
2 | 9nn 9016 | . . . . . 6 ⊢ 9 ∈ ℕ | |
3 | 1nn 8859 | . . . . . 6 ⊢ 1 ∈ ℕ | |
4 | nnaddcl 8868 | . . . . . 6 ⊢ ((9 ∈ ℕ ∧ 1 ∈ ℕ) → (9 + 1) ∈ ℕ) | |
5 | 2, 3, 4 | mp2an 423 | . . . . 5 ⊢ (9 + 1) ∈ ℕ |
6 | 5 | nncni 8858 | . . . 4 ⊢ (9 + 1) ∈ ℂ |
7 | 6 | mulid1i 7892 | . . 3 ⊢ ((9 + 1) · 1) = (9 + 1) |
8 | 7 | oveq1i 5846 | . 2 ⊢ (((9 + 1) · 1) + 0) = ((9 + 1) + 0) |
9 | 6 | addid1i 8031 | . 2 ⊢ ((9 + 1) + 0) = (9 + 1) |
10 | 1, 8, 9 | 3eqtrri 2190 | 1 ⊢ (9 + 1) = ;10 |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∈ wcel 2135 (class class class)co 5836 0cc0 7744 1c1 7745 + caddc 7747 · cmul 7749 ℕcn 8848 9c9 8906 ;cdc 9313 |
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-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 ax-sep 4094 ax-cnex 7835 ax-resscn 7836 ax-1cn 7837 ax-1re 7838 ax-icn 7839 ax-addcl 7840 ax-addrcl 7841 ax-mulcl 7842 ax-mulcom 7845 ax-addass 7846 ax-mulass 7847 ax-distr 7848 ax-1rid 7851 ax-0id 7852 ax-cnre 7855 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-int 3819 df-br 3977 df-iota 5147 df-fv 5190 df-ov 5839 df-inn 8849 df-2 8907 df-3 8908 df-4 8909 df-5 8910 df-6 8911 df-7 8912 df-8 8913 df-9 8914 df-dec 9314 |
This theorem is referenced by: dfdec10 9316 10nn 9328 le9lt10 9339 decsucc 9353 5p5e10 9383 6p4e10 9384 7p3e10 9387 8p2e10 9392 9p2e11 9399 10m1e9 9408 9lt10 9443 sq10e99m1 10615 3dvdsdec 11787 3dvds2dec 11788 |
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