![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > qrevaddcl | GIF version |
Description: Reverse closure law for addition of rationals. (Contributed by NM, 2-Aug-2004.) |
Ref | Expression |
---|---|
qrevaddcl | ⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qcn 9013 | . . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ) | |
2 | pncan 7590 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) | |
3 | 1, 2 | sylan2 280 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℚ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
4 | 3 | ancoms 264 | . . . . . . 7 ⊢ ((𝐵 ∈ ℚ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
5 | 4 | adantr 270 | . . . . . 6 ⊢ (((𝐵 ∈ ℚ ∧ 𝐴 ∈ ℂ) ∧ (𝐴 + 𝐵) ∈ ℚ) → ((𝐴 + 𝐵) − 𝐵) = 𝐴) |
6 | qsubcl 9017 | . . . . . . . 8 ⊢ (((𝐴 + 𝐵) ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴 + 𝐵) − 𝐵) ∈ ℚ) | |
7 | 6 | ancoms 264 | . . . . . . 7 ⊢ ((𝐵 ∈ ℚ ∧ (𝐴 + 𝐵) ∈ ℚ) → ((𝐴 + 𝐵) − 𝐵) ∈ ℚ) |
8 | 7 | adantlr 461 | . . . . . 6 ⊢ (((𝐵 ∈ ℚ ∧ 𝐴 ∈ ℂ) ∧ (𝐴 + 𝐵) ∈ ℚ) → ((𝐴 + 𝐵) − 𝐵) ∈ ℚ) |
9 | 5, 8 | eqeltrrd 2160 | . . . . 5 ⊢ (((𝐵 ∈ ℚ ∧ 𝐴 ∈ ℂ) ∧ (𝐴 + 𝐵) ∈ ℚ) → 𝐴 ∈ ℚ) |
10 | 9 | ex 113 | . . . 4 ⊢ ((𝐵 ∈ ℚ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐵) ∈ ℚ → 𝐴 ∈ ℚ)) |
11 | qaddcl 9014 | . . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 + 𝐵) ∈ ℚ) | |
12 | 11 | expcom 114 | . . . . 5 ⊢ (𝐵 ∈ ℚ → (𝐴 ∈ ℚ → (𝐴 + 𝐵) ∈ ℚ)) |
13 | 12 | adantr 270 | . . . 4 ⊢ ((𝐵 ∈ ℚ ∧ 𝐴 ∈ ℂ) → (𝐴 ∈ ℚ → (𝐴 + 𝐵) ∈ ℚ)) |
14 | 10, 13 | impbid 127 | . . 3 ⊢ ((𝐵 ∈ ℚ ∧ 𝐴 ∈ ℂ) → ((𝐴 + 𝐵) ∈ ℚ ↔ 𝐴 ∈ ℚ)) |
15 | 14 | pm5.32da 440 | . 2 ⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ (𝐴 ∈ ℂ ∧ 𝐴 ∈ ℚ))) |
16 | qcn 9013 | . . 3 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ) | |
17 | 16 | pm4.71ri 384 | . 2 ⊢ (𝐴 ∈ ℚ ↔ (𝐴 ∈ ℂ ∧ 𝐴 ∈ ℚ)) |
18 | 15, 17 | syl6bbr 196 | 1 ⊢ (𝐵 ∈ ℚ → ((𝐴 ∈ ℂ ∧ (𝐴 + 𝐵) ∈ ℚ) ↔ 𝐴 ∈ ℚ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1285 ∈ wcel 1434 (class class class)co 5590 ℂcc 7250 + caddc 7255 − cmin 7555 ℚcq 8998 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-mulrcl 7346 ax-addcom 7347 ax-mulcom 7348 ax-addass 7349 ax-mulass 7350 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-1rid 7354 ax-0id 7355 ax-rnegex 7356 ax-precex 7357 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-ltwlin 7360 ax-pre-lttrn 7361 ax-pre-apti 7362 ax-pre-ltadd 7363 ax-pre-mulgt0 7364 ax-pre-mulext 7365 |
This theorem depends on definitions: df-bi 115 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4083 df-po 4086 df-iso 4087 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-rn 4411 df-res 4412 df-ima 4413 df-iota 4933 df-fun 4970 df-fn 4971 df-f 4972 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-1st 5845 df-2nd 5846 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 df-sub 7557 df-neg 7558 df-reap 7951 df-ap 7958 df-div 8037 df-inn 8316 df-n0 8565 df-z 8646 df-q 8999 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |