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Mirrors > Home > MPE Home > Th. List > issubmndb | Structured version Visualization version GIF version |
Description: The submonoid predicate. Analogous to issubg 18275. (Contributed by AV, 1-Feb-2024.) |
Ref | Expression |
---|---|
issubmndb.b | ⊢ 𝐵 = (Base‘𝐺) |
issubmndb.z | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
issubmndb | ⊢ (𝑆 ∈ (SubMnd‘𝐺) ↔ ((𝐺 ∈ Mnd ∧ (𝐺 ↾s 𝑆) ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubmndb.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | issubmndb.z | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
3 | eqid 2820 | . . . . 5 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
4 | 1, 2, 3 | issubm2 17965 | . . . 4 ⊢ (𝐺 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝐺) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (𝐺 ↾s 𝑆) ∈ Mnd))) |
5 | 3anrot 1095 | . . . . 5 ⊢ (((𝐺 ↾s 𝑆) ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆) ↔ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (𝐺 ↾s 𝑆) ∈ Mnd)) | |
6 | 3anass 1090 | . . . . 5 ⊢ (((𝐺 ↾s 𝑆) ∈ Mnd ∧ 𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆) ↔ ((𝐺 ↾s 𝑆) ∈ Mnd ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆))) | |
7 | 5, 6 | bitr3i 279 | . . . 4 ⊢ ((𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆 ∧ (𝐺 ↾s 𝑆) ∈ Mnd) ↔ ((𝐺 ↾s 𝑆) ∈ Mnd ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆))) |
8 | 4, 7 | syl6bb 289 | . . 3 ⊢ (𝐺 ∈ Mnd → (𝑆 ∈ (SubMnd‘𝐺) ↔ ((𝐺 ↾s 𝑆) ∈ Mnd ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆)))) |
9 | 8 | pm5.32i 577 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑆 ∈ (SubMnd‘𝐺)) ↔ (𝐺 ∈ Mnd ∧ ((𝐺 ↾s 𝑆) ∈ Mnd ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆)))) |
10 | submrcl 17963 | . . 3 ⊢ (𝑆 ∈ (SubMnd‘𝐺) → 𝐺 ∈ Mnd) | |
11 | 10 | pm4.71ri 563 | . 2 ⊢ (𝑆 ∈ (SubMnd‘𝐺) ↔ (𝐺 ∈ Mnd ∧ 𝑆 ∈ (SubMnd‘𝐺))) |
12 | anass 471 | . 2 ⊢ (((𝐺 ∈ Mnd ∧ (𝐺 ↾s 𝑆) ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆)) ↔ (𝐺 ∈ Mnd ∧ ((𝐺 ↾s 𝑆) ∈ Mnd ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆)))) | |
13 | 9, 11, 12 | 3bitr4i 305 | 1 ⊢ (𝑆 ∈ (SubMnd‘𝐺) ↔ ((𝐺 ∈ Mnd ∧ (𝐺 ↾s 𝑆) ∈ Mnd) ∧ (𝑆 ⊆ 𝐵 ∧ 0 ∈ 𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1536 ∈ wcel 2113 ⊆ wss 3933 ‘cfv 6352 (class class class)co 7153 Basecbs 16479 ↾s cress 16480 0gc0g 16709 Mndcmnd 17907 SubMndcsubmnd 17951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5200 ax-nul 5207 ax-pow 5263 ax-pr 5327 ax-un 7458 ax-cnex 10590 ax-resscn 10591 ax-1cn 10592 ax-icn 10593 ax-addcl 10594 ax-addrcl 10595 ax-mulcl 10596 ax-mulrcl 10597 ax-mulcom 10598 ax-addass 10599 ax-mulass 10600 ax-distr 10601 ax-i2m1 10602 ax-1ne0 10603 ax-1rid 10604 ax-rnegex 10605 ax-rrecex 10606 ax-cnre 10607 ax-pre-lttri 10608 ax-pre-lttrn 10609 ax-pre-ltadd 10610 ax-pre-mulgt0 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3495 df-sbc 3771 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4465 df-pw 4538 df-sn 4565 df-pr 4567 df-tp 4569 df-op 4571 df-uni 4836 df-iun 4918 df-br 5064 df-opab 5126 df-mpt 5144 df-tr 5170 df-id 5457 df-eprel 5462 df-po 5471 df-so 5472 df-fr 5511 df-we 5513 df-xp 5558 df-rel 5559 df-cnv 5560 df-co 5561 df-dm 5562 df-rn 5563 df-res 5564 df-ima 5565 df-pred 6145 df-ord 6191 df-on 6192 df-lim 6193 df-suc 6194 df-iota 6311 df-fun 6354 df-fn 6355 df-f 6356 df-f1 6357 df-fo 6358 df-f1o 6359 df-fv 6360 df-riota 7111 df-ov 7156 df-oprab 7157 df-mpo 7158 df-om 7578 df-wrecs 7944 df-recs 8005 df-rdg 8043 df-er 8286 df-en 8507 df-dom 8508 df-sdom 8509 df-pnf 10674 df-mnf 10675 df-xr 10676 df-ltxr 10677 df-le 10678 df-sub 10869 df-neg 10870 df-nn 11636 df-2 11698 df-ndx 16482 df-slot 16483 df-base 16485 df-sets 16486 df-ress 16487 df-plusg 16574 df-0g 16711 df-mgm 17848 df-sgrp 17897 df-mnd 17908 df-submnd 17953 |
This theorem is referenced by: resmndismnd 17969 nsmndex1 18074 symgsubmefmndALT 18527 |
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