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Mirrors > Home > MPE Home > Th. List > Mathboxes > resubeqsub | Structured version Visualization version GIF version |
Description: Equivalence between real subtraction and subtraction. (Contributed by SN, 5-May-2024.) |
Ref | Expression |
---|---|
resubeqsub | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (𝐴 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-resscn 11209 | . . . 4 ⊢ ℝ ⊆ ℂ | |
2 | resubeu 42383 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ∃!𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) | |
3 | reurex 3381 | . . . . 5 ⊢ (∃!𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴 → ∃𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ∃𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) |
5 | recn 11242 | . . . . 5 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℂ) | |
6 | recn 11242 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
7 | sn-subeu 42432 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) | |
8 | 5, 6, 7 | syl2an 596 | . . . 4 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) |
9 | riotass 7418 | . . . 4 ⊢ ((ℝ ⊆ ℂ ∧ ∃𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴 ∧ ∃!𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴) → (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)) | |
10 | 1, 4, 8, 9 | mp3an2i 1465 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ) → (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)) |
11 | 10 | ancoms 458 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)) |
12 | resubval 42373 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴)) | |
13 | subval 11496 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)) | |
14 | 6, 5, 13 | syl2an 596 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 − 𝐵) = (℩𝑥 ∈ ℂ (𝐵 + 𝑥) = 𝐴)) |
15 | 11, 12, 14 | 3eqtr4d 2784 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (𝐴 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∃wrex 3067 ∃!wreu 3375 ⊆ wss 3962 ℩crio 7386 (class class class)co 7430 ℂcc 11150 ℝcr 11151 + caddc 11155 − cmin 11489 −ℝ cresub 42371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-po 5596 df-so 5597 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-ltxr 11297 df-sub 11491 df-2 12326 df-3 12327 df-resub 42372 |
This theorem is referenced by: subresre 42436 reelznn0nn 42455 |
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