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Mirrors > Home > MPE Home > Th. List > 9p4e13 | Structured version Visualization version GIF version |
Description: 9 + 4 = 13. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
9p4e13 | ⊢ (9 + 4) = ;13 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 9nn0 11769 | . 2 ⊢ 9 ∈ ℕ0 | |
2 | 3nn0 11763 | . 2 ⊢ 3 ∈ ℕ0 | |
3 | 2nn0 11762 | . 2 ⊢ 2 ∈ ℕ0 | |
4 | df-4 11550 | . 2 ⊢ 4 = (3 + 1) | |
5 | df-3 11549 | . 2 ⊢ 3 = (2 + 1) | |
6 | 9p3e12 12036 | . 2 ⊢ (9 + 3) = ;12 | |
7 | 1, 2, 3, 4, 5, 6 | 6p5lem 12018 | 1 ⊢ (9 + 4) = ;13 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1522 (class class class)co 7016 1c1 10384 + caddc 10386 2c2 11540 3c3 11541 4c4 11542 9c9 11547 ;cdc 11947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-ov 7019 df-om 7437 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-ltxr 10526 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-dec 11948 |
This theorem is referenced by: 9p5e14 12038 9t7e63 12075 43prm 16284 83prm 16285 163prm 16287 2503lem2 16300 2503lem3 16301 log2ub 25209 hgt750lem2 31540 139prmALT 43241 |
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