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Mirrors > Home > ILE Home > Th. List > strle3g | GIF version |
Description: Make a structure from a triple. (Contributed by Mario Carneiro, 29-Aug-2015.) |
Ref | Expression |
---|---|
strle1.i | ⊢ 𝐼 ∈ ℕ |
strle1.a | ⊢ 𝐴 = 𝐼 |
strle2.j | ⊢ 𝐼 < 𝐽 |
strle2.k | ⊢ 𝐽 ∈ ℕ |
strle2.b | ⊢ 𝐵 = 𝐽 |
strle3.k | ⊢ 𝐽 < 𝐾 |
strle3.l | ⊢ 𝐾 ∈ ℕ |
strle3.c | ⊢ 𝐶 = 𝐾 |
Ref | Expression |
---|---|
strle3g | ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑃) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉} Struct 〈𝐼, 𝐾〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-tp 3501 | . 2 ⊢ {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉} = ({〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} ∪ {〈𝐶, 𝑍〉}) | |
2 | strle1.i | . . . . 5 ⊢ 𝐼 ∈ ℕ | |
3 | strle1.a | . . . . 5 ⊢ 𝐴 = 𝐼 | |
4 | strle2.j | . . . . 5 ⊢ 𝐼 < 𝐽 | |
5 | strle2.k | . . . . 5 ⊢ 𝐽 ∈ ℕ | |
6 | strle2.b | . . . . 5 ⊢ 𝐵 = 𝐽 | |
7 | 2, 3, 4, 5, 6 | strle2g 11893 | . . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} Struct 〈𝐼, 𝐽〉) |
8 | 7 | 3adant3 984 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑃) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} Struct 〈𝐼, 𝐽〉) |
9 | strle3.l | . . . . 5 ⊢ 𝐾 ∈ ℕ | |
10 | strle3.c | . . . . 5 ⊢ 𝐶 = 𝐾 | |
11 | 9, 10 | strle1g 11892 | . . . 4 ⊢ (𝑍 ∈ 𝑃 → {〈𝐶, 𝑍〉} Struct 〈𝐾, 𝐾〉) |
12 | 11 | 3ad2ant3 987 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑃) → {〈𝐶, 𝑍〉} Struct 〈𝐾, 𝐾〉) |
13 | strle3.k | . . . 4 ⊢ 𝐽 < 𝐾 | |
14 | 13 | a1i 9 | . . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑃) → 𝐽 < 𝐾) |
15 | 8, 12, 14 | strleund 11890 | . 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑃) → ({〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉} ∪ {〈𝐶, 𝑍〉}) Struct 〈𝐼, 𝐾〉) |
16 | 1, 15 | eqbrtrid 3928 | 1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑊 ∧ 𝑍 ∈ 𝑃) → {〈𝐴, 𝑋〉, 〈𝐵, 𝑌〉, 〈𝐶, 𝑍〉} Struct 〈𝐼, 𝐾〉) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 945 = wceq 1314 ∈ wcel 1463 ∪ cun 3035 {csn 3493 {cpr 3494 {ctp 3495 〈cop 3496 class class class wbr 3895 < clt 7724 ℕcn 8630 Struct cstr 11798 |
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 586 ax-in2 587 ax-io 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-13 1474 ax-14 1475 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 ax-sep 4006 ax-pow 4058 ax-pr 4091 ax-un 4315 ax-setind 4412 ax-cnex 7636 ax-resscn 7637 ax-1cn 7638 ax-1re 7639 ax-icn 7640 ax-addcl 7641 ax-addrcl 7642 ax-mulcl 7643 ax-addcom 7645 ax-addass 7647 ax-distr 7649 ax-i2m1 7650 ax-0lt1 7651 ax-0id 7653 ax-rnegex 7654 ax-cnre 7656 ax-pre-ltirr 7657 ax-pre-ltwlin 7658 ax-pre-lttrn 7659 ax-pre-apti 7660 ax-pre-ltadd 7661 |
This theorem depends on definitions: df-bi 116 df-3or 946 df-3an 947 df-tru 1317 df-fal 1320 df-nf 1420 df-sb 1719 df-eu 1978 df-mo 1979 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-ne 2283 df-nel 2378 df-ral 2395 df-rex 2396 df-reu 2397 df-rab 2399 df-v 2659 df-sbc 2879 df-dif 3039 df-un 3041 df-in 3043 df-ss 3050 df-nul 3330 df-pw 3478 df-sn 3499 df-pr 3500 df-tp 3501 df-op 3502 df-uni 3703 df-int 3738 df-br 3896 df-opab 3950 df-mpt 3951 df-id 4175 df-xp 4505 df-rel 4506 df-cnv 4507 df-co 4508 df-dm 4509 df-rn 4510 df-res 4511 df-ima 4512 df-iota 5046 df-fun 5083 df-fn 5084 df-f 5085 df-fv 5089 df-riota 5684 df-ov 5731 df-oprab 5732 df-mpo 5733 df-pnf 7726 df-mnf 7727 df-xr 7728 df-ltxr 7729 df-le 7730 df-sub 7858 df-neg 7859 df-inn 8631 df-n0 8882 df-z 8959 df-uz 9229 df-fz 9684 df-struct 11804 |
This theorem is referenced by: rngstrg 11917 lmodstrd 11935 ipsstrd 11943 topgrpstrd 11953 |
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