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Mirrors > Home > MPE Home > Th. List > 4m1e3 | Structured version Visualization version GIF version |
Description: 4 - 1 = 3. (Contributed by AV, 8-Feb-2021.) (Proof shortened by AV, 6-Sep-2021.) |
Ref | Expression |
---|---|
4m1e3 | ⊢ (4 − 1) = 3 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3cn 12231 | . 2 ⊢ 3 ∈ ℂ | |
2 | ax-1cn 11106 | . 2 ⊢ 1 ∈ ℂ | |
3 | df-4 12215 | . 2 ⊢ 4 = (3 + 1) | |
4 | 1, 2, 3 | mvrraddi 11415 | 1 ⊢ (4 − 1) = 3 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 (class class class)co 7354 1c1 11049 − cmin 11382 3c3 12206 4c4 12207 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-mpt 5188 df-id 5530 df-po 5544 df-so 5545 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-er 8645 df-en 8881 df-dom 8882 df-sdom 8883 df-pnf 11188 df-mnf 11189 df-ltxr 11191 df-sub 11384 df-2 12213 df-3 12214 df-4 12215 |
This theorem is referenced by: fzo0to42pr 13656 fzo1to4tp 13657 4bc3eq4 14225 lsws4 14792 bpoly4 15939 prmo4 16997 iblitg 25129 sincos6thpi 25868 ang180lem2 26156 log2ub 26295 ppiub 26548 bclbnd 26624 3pthd 29016 hgt750lemd 33152 lcm4un 40462 aks4d1p1p5 40521 fmtno4sqrt 45733 m2prm 45753 lighneallem2 45768 4fppr1 45897 fpprel2 45903 |
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