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Mirrors > Home > MPE Home > Th. List > Mathboxes > assraddsubi | Structured version Visualization version GIF version |
Description: Associate RHS addition-subtraction. (Contributed by David A. Wheeler, 11-Oct-2018.) |
Ref | Expression |
---|---|
assraddsubi.1 | ⊢ 𝐵 ∈ ℂ |
assraddsubi.2 | ⊢ 𝐶 ∈ ℂ |
assraddsubi.3 | ⊢ 𝐷 ∈ ℂ |
assraddsubi.4 | ⊢ 𝐴 = ((𝐵 + 𝐶) − 𝐷) |
Ref | Expression |
---|---|
assraddsubi | ⊢ 𝐴 = (𝐵 + (𝐶 − 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | assraddsubi.4 | . 2 ⊢ 𝐴 = ((𝐵 + 𝐶) − 𝐷) | |
2 | assraddsubi.1 | . . 3 ⊢ 𝐵 ∈ ℂ | |
3 | assraddsubi.2 | . . 3 ⊢ 𝐶 ∈ ℂ | |
4 | assraddsubi.3 | . . 3 ⊢ 𝐷 ∈ ℂ | |
5 | 2, 3, 4 | addsubassi 10776 | . 2 ⊢ ((𝐵 + 𝐶) − 𝐷) = (𝐵 + (𝐶 − 𝐷)) |
6 | 1, 5 | eqtri 2795 | 1 ⊢ 𝐴 = (𝐵 + (𝐶 − 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1508 ∈ wcel 2051 (class class class)co 6974 ℂcc 10331 + caddc 10336 − cmin 10668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-resscn 10390 ax-1cn 10391 ax-icn 10392 ax-addcl 10393 ax-addrcl 10394 ax-mulcl 10395 ax-mulrcl 10396 ax-mulcom 10397 ax-addass 10398 ax-mulass 10399 ax-distr 10400 ax-i2m1 10401 ax-1ne0 10402 ax-1rid 10403 ax-rnegex 10404 ax-rrecex 10405 ax-cnre 10406 ax-pre-lttri 10407 ax-pre-lttrn 10408 ax-pre-ltadd 10409 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-nel 3067 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-po 5322 df-so 5323 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-er 8087 df-en 8305 df-dom 8306 df-sdom 8307 df-pnf 10474 df-mnf 10475 df-ltxr 10477 df-sub 10670 |
This theorem is referenced by: i2linesi 44280 |
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