![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > cnm2m1cnm3 | GIF version |
Description: Subtracting 2 and afterwards 1 from a number results in the difference between the number and 3. (Contributed by Alexander van der Vekens, 16-Sep-2018.) |
Ref | Expression |
---|---|
cnm2m1cnm3 | ⊢ (A ∈ ℂ → ((A − 2) − 1) = (A − 3)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . . 3 ⊢ (A ∈ ℂ → A ∈ ℂ) | |
2 | 2cnd 7768 | . . 3 ⊢ (A ∈ ℂ → 2 ∈ ℂ) | |
3 | 1cnd 6841 | . . 3 ⊢ (A ∈ ℂ → 1 ∈ ℂ) | |
4 | 1, 2, 3 | subsub4d 7149 | . 2 ⊢ (A ∈ ℂ → ((A − 2) − 1) = (A − (2 + 1))) |
5 | 2p1e3 7821 | . . . 4 ⊢ (2 + 1) = 3 | |
6 | 5 | a1i 9 | . . 3 ⊢ (A ∈ ℂ → (2 + 1) = 3) |
7 | 6 | oveq2d 5471 | . 2 ⊢ (A ∈ ℂ → (A − (2 + 1)) = (A − 3)) |
8 | 4, 7 | eqtrd 2069 | 1 ⊢ (A ∈ ℂ → ((A − 2) − 1) = (A − 3)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1242 ∈ wcel 1390 (class class class)co 5455 ℂcc 6709 1c1 6712 + caddc 6714 − cmin 6979 2c2 7744 3c3 7745 |
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-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-setind 4220 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-addcom 6783 ax-addass 6785 ax-distr 6787 ax-i2m1 6788 ax-0id 6791 ax-rnegex 6792 ax-cnre 6794 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-iota 4810 df-fun 4847 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-sub 6981 df-2 7753 df-3 7754 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |