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Mirrors > Home > MPE Home > Th. List > Mathboxes > repncan3 | Structured version Visualization version GIF version |
Description: Addition and subtraction of equals. Based on pncan3 10886. (Contributed by Steven Nguyen, 8-Jan-2023.) |
Ref | Expression |
---|---|
repncan3 | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (𝐵 −ℝ 𝐴)) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rersubcl 39199 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐵 −ℝ 𝐴) ∈ ℝ) | |
2 | eqid 2819 | . . . 4 ⊢ (𝐵 −ℝ 𝐴) = (𝐵 −ℝ 𝐴) | |
3 | resubadd 39200 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐵 −ℝ 𝐴) ∈ ℝ) → ((𝐵 −ℝ 𝐴) = (𝐵 −ℝ 𝐴) ↔ (𝐴 + (𝐵 −ℝ 𝐴)) = 𝐵)) | |
4 | 2, 3 | mpbii 235 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ (𝐵 −ℝ 𝐴) ∈ ℝ) → (𝐴 + (𝐵 −ℝ 𝐴)) = 𝐵) |
5 | 1, 4 | mpd3an3 1456 | . 2 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (𝐴 + (𝐵 −ℝ 𝐴)) = 𝐵) |
6 | 5 | ancoms 461 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 + (𝐵 −ℝ 𝐴)) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∈ wcel 2108 (class class class)co 7148 ℝcr 10528 + caddc 10532 −ℝ cresub 39186 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-resscn 10586 ax-addrcl 10590 ax-addass 10594 ax-rnegex 10600 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-pnf 10669 df-mnf 10670 df-ltxr 10672 df-resub 39187 |
This theorem is referenced by: readdsub 39205 reltsub1 39207 resubcan2 39209 resubsub4 39210 rennncan2 39211 renpncan3 39212 resubdi 39217 re1m1e0m0 39218 renegneg 39232 |
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