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Mirrors > Home > ILE Home > Th. List > genpassg | GIF version |
Description: Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.) |
Ref | Expression |
---|---|
genpelvl.1 | ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ 〈{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}〉) |
genpelvl.2 | ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) |
genpassg.4 | ⊢ dom 𝐹 = (P × P) |
genpassg.5 | ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P) |
genpassg.6 | ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ))) |
Ref | Expression |
---|---|
genpassg | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | genpelvl.1 | . . 3 ⊢ 𝐹 = (𝑤 ∈ P, 𝑣 ∈ P ↦ 〈{𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (1st ‘𝑤) ∧ 𝑧 ∈ (1st ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}, {𝑥 ∈ Q ∣ ∃𝑦 ∈ Q ∃𝑧 ∈ Q (𝑦 ∈ (2nd ‘𝑤) ∧ 𝑧 ∈ (2nd ‘𝑣) ∧ 𝑥 = (𝑦𝐺𝑧))}〉) | |
2 | genpelvl.2 | . . 3 ⊢ ((𝑦 ∈ Q ∧ 𝑧 ∈ Q) → (𝑦𝐺𝑧) ∈ Q) | |
3 | genpassg.4 | . . 3 ⊢ dom 𝐹 = (P × P) | |
4 | genpassg.5 | . . 3 ⊢ ((𝑓 ∈ P ∧ 𝑔 ∈ P) → (𝑓𝐹𝑔) ∈ P) | |
5 | genpassg.6 | . . 3 ⊢ ((𝑓 ∈ Q ∧ 𝑔 ∈ Q ∧ ℎ ∈ Q) → ((𝑓𝐺𝑔)𝐺ℎ) = (𝑓𝐺(𝑔𝐺ℎ))) | |
6 | 1, 2, 3, 4, 5 | genpassl 7180 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶)))) |
7 | 1, 2, 3, 4, 5 | genpassu 7181 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶)))) |
8 | 4 | caovcl 5837 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴𝐹𝐵) ∈ P) |
9 | 4 | caovcl 5837 | . . . . 5 ⊢ (((𝐴𝐹𝐵) ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) ∈ P) |
10 | 8, 9 | sylan 278 | . . . 4 ⊢ (((𝐴 ∈ P ∧ 𝐵 ∈ P) ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) ∈ P) |
11 | 10 | 3impa 1141 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) ∈ P) |
12 | 4 | caovcl 5837 | . . . . 5 ⊢ ((𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐵𝐹𝐶) ∈ P) |
13 | 4 | caovcl 5837 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ (𝐵𝐹𝐶) ∈ P) → (𝐴𝐹(𝐵𝐹𝐶)) ∈ P) |
14 | 12, 13 | sylan2 281 | . . . 4 ⊢ ((𝐴 ∈ P ∧ (𝐵 ∈ P ∧ 𝐶 ∈ P)) → (𝐴𝐹(𝐵𝐹𝐶)) ∈ P) |
15 | 14 | 3impb 1142 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (𝐴𝐹(𝐵𝐹𝐶)) ∈ P) |
16 | preqlu 7128 | . . 3 ⊢ ((((𝐴𝐹𝐵)𝐹𝐶) ∈ P ∧ (𝐴𝐹(𝐵𝐹𝐶)) ∈ P) → (((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) ↔ ((1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ∧ (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶)))))) | |
17 | 11, 15, 16 | syl2anc 404 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → (((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶)) ↔ ((1st ‘((𝐴𝐹𝐵)𝐹𝐶)) = (1st ‘(𝐴𝐹(𝐵𝐹𝐶))) ∧ (2nd ‘((𝐴𝐹𝐵)𝐹𝐶)) = (2nd ‘(𝐴𝐹(𝐵𝐹𝐶)))))) |
18 | 6, 7, 17 | mpbir2and 893 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ 𝐶 ∈ P) → ((𝐴𝐹𝐵)𝐹𝐶) = (𝐴𝐹(𝐵𝐹𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 ∃wrex 2371 {crab 2374 〈cop 3469 × cxp 4465 dom cdm 4467 ‘cfv 5049 (class class class)co 5690 ↦ cmpt2 5692 1st c1st 5947 2nd c2nd 5948 Qcnq 6936 Pcnp 6947 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-coll 3975 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-reu 2377 df-rab 2379 df-v 2635 df-sbc 2855 df-csb 2948 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-iun 3754 df-br 3868 df-opab 3922 df-mpt 3923 df-id 4144 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-1st 5949 df-2nd 5950 df-qs 6338 df-ni 6960 df-nqqs 7004 df-inp 7122 |
This theorem is referenced by: addassprg 7235 mulassprg 7237 |
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