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Mirrors > Home > ILE Home > Th. List > negsubdii | GIF version |
Description: Distribution of negative over subtraction. (Contributed by NM, 6-Aug-1999.) |
Ref | Expression |
---|---|
negidi.1 | ⊢ 𝐴 ∈ ℂ |
pncan3i.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
negsubdii | ⊢ -(𝐴 − 𝐵) = (-𝐴 + 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negidi.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
2 | pncan3i.2 | . . . 4 ⊢ 𝐵 ∈ ℂ | |
3 | 2 | negcli 8180 | . . 3 ⊢ -𝐵 ∈ ℂ |
4 | 1, 3 | negdii 8196 | . 2 ⊢ -(𝐴 + -𝐵) = (-𝐴 + --𝐵) |
5 | 1, 2 | negsubi 8190 | . . 3 ⊢ (𝐴 + -𝐵) = (𝐴 − 𝐵) |
6 | 5 | negeqi 8106 | . 2 ⊢ -(𝐴 + -𝐵) = -(𝐴 − 𝐵) |
7 | 2 | negnegi 8182 | . . 3 ⊢ --𝐵 = 𝐵 |
8 | 7 | oveq2i 5862 | . 2 ⊢ (-𝐴 + --𝐵) = (-𝐴 + 𝐵) |
9 | 4, 6, 8 | 3eqtr3i 2199 | 1 ⊢ -(𝐴 − 𝐵) = (-𝐴 + 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 (class class class)co 5851 ℂcc 7765 + caddc 7770 − cmin 8083 -cneg 8084 |
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-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-setind 4519 ax-resscn 7859 ax-1cn 7860 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-addcom 7867 ax-addass 7869 ax-distr 7871 ax-i2m1 7872 ax-0id 7875 ax-rnegex 7876 ax-cnre 7878 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-sub 8085 df-neg 8086 |
This theorem is referenced by: negsubdi2i 8198 resqrexlemover 10967 |
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