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Mirrors > Home > ILE Home > Th. List > addvalex | GIF version |
Description: Existence of a sum. This is dependent on how we define + so once we proceed to real number axioms we will replace it with theorems such as addcl 7745. (Contributed by Jim Kingdon, 14-Jul-2021.) |
Ref | Expression |
---|---|
addvalex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 + 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ov 5777 | . 2 ⊢ (𝐴 + 𝐵) = ( + ‘〈𝐴, 𝐵〉) | |
2 | df-nr 7535 | . . . . 5 ⊢ R = ((P × P) / ~R ) | |
3 | npex 7281 | . . . . . . 7 ⊢ P ∈ V | |
4 | 3, 3 | xpex 4654 | . . . . . 6 ⊢ (P × P) ∈ V |
5 | 4 | qsex 6486 | . . . . 5 ⊢ ((P × P) / ~R ) ∈ V |
6 | 2, 5 | eqeltri 2212 | . . . 4 ⊢ R ∈ V |
7 | df-add 7631 | . . . . 5 ⊢ + = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} | |
8 | df-c 7626 | . . . . . . . . 9 ⊢ ℂ = (R × R) | |
9 | 8 | eleq2i 2206 | . . . . . . . 8 ⊢ (𝑥 ∈ ℂ ↔ 𝑥 ∈ (R × R)) |
10 | 8 | eleq2i 2206 | . . . . . . . 8 ⊢ (𝑦 ∈ ℂ ↔ 𝑦 ∈ (R × R)) |
11 | 9, 10 | anbi12i 455 | . . . . . . 7 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ↔ (𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R))) |
12 | 11 | anbi1i 453 | . . . . . 6 ⊢ (((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉)) ↔ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))) |
13 | 12 | oprabbii 5826 | . . . . 5 ⊢ {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} |
14 | 7, 13 | eqtri 2160 | . . . 4 ⊢ + = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ (R × R) ∧ 𝑦 ∈ (R × R)) ∧ ∃𝑤∃𝑣∃𝑢∃𝑓((𝑥 = 〈𝑤, 𝑣〉 ∧ 𝑦 = 〈𝑢, 𝑓〉) ∧ 𝑧 = 〈(𝑤 +R 𝑢), (𝑣 +R 𝑓)〉))} |
15 | 6, 14 | oprabex3 6027 | . . 3 ⊢ + ∈ V |
16 | opexg 4150 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 〈𝐴, 𝐵〉 ∈ V) | |
17 | fvexg 5440 | . . 3 ⊢ (( + ∈ V ∧ 〈𝐴, 𝐵〉 ∈ V) → ( + ‘〈𝐴, 𝐵〉) ∈ V) | |
18 | 15, 16, 17 | sylancr 410 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ( + ‘〈𝐴, 𝐵〉) ∈ V) |
19 | 1, 18 | eqeltrid 2226 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 + 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∃wex 1468 ∈ wcel 1480 Vcvv 2686 〈cop 3530 × cxp 4537 ‘cfv 5123 (class class class)co 5774 {coprab 5775 / cqs 6428 Pcnp 7099 ~R cer 7104 Rcnr 7105 +R cplr 7109 ℂcc 7618 + caddc 7623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-qs 6435 df-ni 7112 df-nqqs 7156 df-inp 7274 df-nr 7535 df-c 7626 df-add 7631 |
This theorem is referenced by: peano2nnnn 7661 |
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