Mathbox for Steven Nguyen |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > reladdrsub | Structured version Visualization version GIF version |
Description: Move LHS of a sum into RHS of a (real) difference. Version of mvlladdd 11048 with real subtraction. (Contributed by Steven Nguyen, 8-Jan-2023.) |
Ref | Expression |
---|---|
reladdrsub.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
reladdrsub.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
reladdrsub.3 | ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶) |
Ref | Expression |
---|---|
reladdrsub | ⊢ (𝜑 → 𝐵 = (𝐶 −ℝ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reladdrsub.3 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) = 𝐶) | |
2 | reladdrsub.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | reladdrsub.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 2, 3 | readdcld 10667 | . . . 4 ⊢ (𝜑 → (𝐴 + 𝐵) ∈ ℝ) |
5 | 1, 4 | eqeltrrd 2913 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
6 | resubadd 39284 | . . . 4 ⊢ ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐶 −ℝ 𝐴) = 𝐵 ↔ (𝐴 + 𝐵) = 𝐶)) | |
7 | 1, 6 | syl5ibrcom 249 | . . 3 ⊢ (𝜑 → ((𝐶 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 −ℝ 𝐴) = 𝐵)) |
8 | 5, 2, 3, 7 | mp3and 1459 | . 2 ⊢ (𝜑 → (𝐶 −ℝ 𝐴) = 𝐵) |
9 | 8 | eqcomd 2826 | 1 ⊢ (𝜑 → 𝐵 = (𝐶 −ℝ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 (class class class)co 7153 ℝcr 10533 + caddc 10537 −ℝ cresub 39270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-resscn 10591 ax-addrcl 10595 ax-addass 10599 ax-rnegex 10605 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4836 df-br 5064 df-opab 5126 df-mpt 5144 df-id 5457 df-po 5471 df-so 5472 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-pnf 10674 df-mnf 10675 df-ltxr 10677 df-resub 39271 |
This theorem is referenced by: resubsub4 39294 resubidaddid1lem 39299 resubdi 39301 re1m1e0m0 39302 re0m0e0 39307 |
Copyright terms: Public domain | W3C validator |