![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > issubmd | GIF version |
Description: Deduction for proving a submonoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Stefan O'Rear, 5-Sep-2015.) |
Ref | Expression |
---|---|
issubmd.b | ⊢ 𝐵 = (Base‘𝑀) |
issubmd.p | ⊢ + = (+g‘𝑀) |
issubmd.z | ⊢ 0 = (0g‘𝑀) |
issubmd.m | ⊢ (𝜑 → 𝑀 ∈ Mnd) |
issubmd.cz | ⊢ (𝜑 → 𝜒) |
issubmd.cp | ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂) |
issubmd.ch | ⊢ (𝑧 = 0 → (𝜓 ↔ 𝜒)) |
issubmd.th | ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃)) |
issubmd.ta | ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏)) |
issubmd.et | ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂)) |
Ref | Expression |
---|---|
issubmd | ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMnd‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 3240 | . . 3 ⊢ {𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵 | |
2 | 1 | a1i 9 | . 2 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵) |
3 | issubmd.ch | . . 3 ⊢ (𝑧 = 0 → (𝜓 ↔ 𝜒)) | |
4 | issubmd.m | . . . 4 ⊢ (𝜑 → 𝑀 ∈ Mnd) | |
5 | issubmd.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑀) | |
6 | issubmd.z | . . . . 5 ⊢ 0 = (0g‘𝑀) | |
7 | 5, 6 | mndidcl 12710 | . . . 4 ⊢ (𝑀 ∈ Mnd → 0 ∈ 𝐵) |
8 | 4, 7 | syl 14 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) |
9 | issubmd.cz | . . 3 ⊢ (𝜑 → 𝜒) | |
10 | 3, 8, 9 | elrabd 2895 | . 2 ⊢ (𝜑 → 0 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}) |
11 | issubmd.th | . . . . . 6 ⊢ (𝑧 = 𝑥 → (𝜓 ↔ 𝜃)) | |
12 | 11 | elrab 2893 | . . . . 5 ⊢ (𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ (𝑥 ∈ 𝐵 ∧ 𝜃)) |
13 | issubmd.ta | . . . . . 6 ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜏)) | |
14 | 13 | elrab 2893 | . . . . 5 ⊢ (𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ↔ (𝑦 ∈ 𝐵 ∧ 𝜏)) |
15 | 12, 14 | anbi12i 460 | . . . 4 ⊢ ((𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}) ↔ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) |
16 | issubmd.et | . . . . 5 ⊢ (𝑧 = (𝑥 + 𝑦) → (𝜓 ↔ 𝜂)) | |
17 | 4 | adantr 276 | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑀 ∈ Mnd) |
18 | simprll 537 | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑥 ∈ 𝐵) | |
19 | simprrl 539 | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝑦 ∈ 𝐵) | |
20 | issubmd.p | . . . . . . 7 ⊢ + = (+g‘𝑀) | |
21 | 5, 20 | mndcl 12703 | . . . . . 6 ⊢ ((𝑀 ∈ Mnd ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) ∈ 𝐵) |
22 | 17, 18, 19, 21 | syl3anc 1238 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → (𝑥 + 𝑦) ∈ 𝐵) |
23 | an4 586 | . . . . . 6 ⊢ (((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) | |
24 | issubmd.cp | . . . . . 6 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ (𝜃 ∧ 𝜏))) → 𝜂) | |
25 | 23, 24 | sylan2b 287 | . . . . 5 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → 𝜂) |
26 | 16, 22, 25 | elrabd 2895 | . . . 4 ⊢ ((𝜑 ∧ ((𝑥 ∈ 𝐵 ∧ 𝜃) ∧ (𝑦 ∈ 𝐵 ∧ 𝜏))) → (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}) |
27 | 15, 26 | sylan2b 287 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ 𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓})) → (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}) |
28 | 27 | ralrimivva 2559 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}) |
29 | 5, 6, 20 | issubm 12740 | . . 3 ⊢ (𝑀 ∈ Mnd → ({𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMnd‘𝑀) ↔ ({𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵 ∧ 0 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}))) |
30 | 4, 29 | syl 14 | . 2 ⊢ (𝜑 → ({𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMnd‘𝑀) ↔ ({𝑧 ∈ 𝐵 ∣ 𝜓} ⊆ 𝐵 ∧ 0 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} ∧ ∀𝑥 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}∀𝑦 ∈ {𝑧 ∈ 𝐵 ∣ 𝜓} (𝑥 + 𝑦) ∈ {𝑧 ∈ 𝐵 ∣ 𝜓}))) |
31 | 2, 10, 28, 30 | mpbir3and 1180 | 1 ⊢ (𝜑 → {𝑧 ∈ 𝐵 ∣ 𝜓} ∈ (SubMnd‘𝑀)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ∀wral 2455 {crab 2459 ⊆ wss 3129 ‘cfv 5211 (class class class)co 5868 Basecbs 12432 +gcplusg 12505 0gc0g 12640 Mndcmnd 12696 SubMndcsubmnd 12727 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-iota 5173 df-fun 5213 df-fn 5214 df-fv 5219 df-riota 5824 df-ov 5871 df-inn 8896 df-2 8954 df-ndx 12435 df-slot 12436 df-base 12438 df-plusg 12518 df-0g 12642 df-mgm 12654 df-sgrp 12687 df-mnd 12697 df-submnd 12729 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |