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Mirrors > Home > MPE Home > Th. List > 7t5e35 | Structured version Visualization version GIF version |
Description: 7 times 5 equals 35. (Contributed by Mario Carneiro, 19-Apr-2015.) |
Ref | Expression |
---|---|
7t5e35 | ⊢ (7 · 5) = ;35 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 7nn0 12540 | . 2 ⊢ 7 ∈ ℕ0 | |
2 | 4nn0 12537 | . 2 ⊢ 4 ∈ ℕ0 | |
3 | df-5 12324 | . 2 ⊢ 5 = (4 + 1) | |
4 | 7t4e28 12834 | . 2 ⊢ (7 · 4) = ;28 | |
5 | 2nn0 12535 | . . 3 ⊢ 2 ∈ ℕ0 | |
6 | 8nn0 12541 | . . 3 ⊢ 8 ∈ ℕ0 | |
7 | eqid 2726 | . . 3 ⊢ ;28 = ;28 | |
8 | 2p1e3 12400 | . . 3 ⊢ (2 + 1) = 3 | |
9 | 5nn0 12538 | . . 3 ⊢ 5 ∈ ℕ0 | |
10 | 8p7e15 12808 | . . 3 ⊢ (8 + 7) = ;15 | |
11 | 5, 6, 1, 7, 8, 9, 10 | decaddci 12784 | . 2 ⊢ (;28 + 7) = ;35 |
12 | 1, 2, 3, 4, 11 | 4t3lem 12820 | 1 ⊢ (7 · 5) = ;35 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 (class class class)co 7416 · cmul 11154 2c2 12313 3c3 12314 4c4 12315 5c5 12316 7c7 12318 8c8 12319 ;cdc 12723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-pnf 11291 df-mnf 11292 df-ltxr 11294 df-sub 11487 df-nn 12259 df-2 12321 df-3 12322 df-4 12323 df-5 12324 df-6 12325 df-7 12326 df-8 12327 df-9 12328 df-n0 12519 df-dec 12724 |
This theorem is referenced by: 7t6e42 12836 37prm 17118 317prm 17123 log2ublem3 26973 log2ub 26974 235t711 42032 ex-decpmul 42033 257prm 47169 |
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