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Mirrors > Home > MPE Home > Th. List > quselbas | Structured version Visualization version GIF version |
Description: Membership in the base set of a quotient group. (Contributed by AV, 1-Mar-2025.) |
Ref | Expression |
---|---|
quselbas.e | ⊢ ∼ = (𝐺 ~QG 𝑆) |
quselbas.u | ⊢ 𝑈 = (𝐺 /s ∼ ) |
quselbas.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
quselbas | ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (𝑋 ∈ (Base‘𝑈) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | quselbas.u | . . . . . 6 ⊢ 𝑈 = (𝐺 /s ∼ ) | |
2 | 1 | a1i 11 | . . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → 𝑈 = (𝐺 /s ∼ )) |
3 | quselbas.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
4 | 3 | a1i 11 | . . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → 𝐵 = (Base‘𝐺)) |
5 | quselbas.e | . . . . . . 7 ⊢ ∼ = (𝐺 ~QG 𝑆) | |
6 | 5 | ovexi 7460 | . . . . . 6 ⊢ ∼ ∈ V |
7 | 6 | a1i 11 | . . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → ∼ ∈ V) |
8 | simpl 481 | . . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → 𝐺 ∈ 𝑉) | |
9 | 2, 4, 7, 8 | qusbas 17534 | . . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (𝐵 / ∼ ) = (Base‘𝑈)) |
10 | 9 | eqcomd 2734 | . . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (Base‘𝑈) = (𝐵 / ∼ )) |
11 | 10 | eleq2d 2815 | . 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (𝑋 ∈ (Base‘𝑈) ↔ 𝑋 ∈ (𝐵 / ∼ ))) |
12 | elqsg 8793 | . . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑋 ∈ (𝐵 / ∼ ) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ )) | |
13 | 12 | adantl 480 | . 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (𝑋 ∈ (𝐵 / ∼ ) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ )) |
14 | 11, 13 | bitrd 278 | 1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊) → (𝑋 ∈ (Base‘𝑈) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3067 Vcvv 3473 ‘cfv 6553 (class class class)co 7426 [cec 8729 / cqs 8730 Basecbs 17187 /s cqus 17494 ~QG cqg 19084 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-ec 8733 df-qs 8737 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-fz 13525 df-struct 17123 df-slot 17158 df-ndx 17170 df-base 17188 df-plusg 17253 df-mulr 17254 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-imas 17497 df-qus 17498 |
This theorem is referenced by: rngqiprngimfo 21198 |
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