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Theorem quselbasg 14093

Description: Membership in the base set of a quotient group. (Contributed by AV, 1-Mar-2025.)

**Hypotheses**
Ref	Expression
quselbas.e	⊢ ∼ = (𝐺 ~_QG 𝑆)
quselbas.u	⊢ 𝑈 = (𝐺 /_s ∼ )
quselbas.b	⊢ 𝐵 = (Base‘𝐺)

**Assertion**
Ref	Expression
quselbasg	⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → (𝑋 ∈ (Base‘𝑈) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ ))

Distinct variable groups: 𝑥,𝐵 𝑥,𝑋 𝑥, ∼ Allowed substitution hints: 𝑆(𝑥) 𝑈(𝑥) 𝐺(𝑥) 𝑉(𝑥) 𝑊(𝑥) 𝑍(𝑥)

**Proof of Theorem quselbasg**
Step	Hyp	Ref	Expression
1		quselbas.u	. . . . . 6 ⊢ 𝑈 = (𝐺 /_s ∼ )
2	1	a1i 9	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → 𝑈 = (𝐺 /_s ∼ ))
3		quselbas.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4	3	a1i 9	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → 𝐵 = (Base‘𝐺))
5		quselbas.e	. . . . . 6 ⊢ ∼ = (𝐺 ~_QG 𝑆)
6		eqgex 14084	. . . . . . 7 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑆 ∈ 𝑍) → (𝐺 ~_QG 𝑆) ∈ V)
7	6	3adant2 1047	. . . . . 6 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → (𝐺 ~_QG 𝑆) ∈ V)
8	5, 7	eqeltrid 2325	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → ∼ ∈ V)
9		simp1 1028	. . . . 5 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → 𝐺 ∈ 𝑉)
10	2, 4, 8, 9	qusbas 13708	. . . 4 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → (𝐵 / ∼ ) = (Base‘𝑈))
11	10	eqcomd 2244	. . 3 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → (Base‘𝑈) = (𝐵 / ∼ ))
12	11	eleq2d 2308	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → (𝑋 ∈ (Base‘𝑈) ↔ 𝑋 ∈ (𝐵 / ∼ )))
13		elqsg 6863	. . 3 ⊢ (𝑋 ∈ 𝑊 → (𝑋 ∈ (𝐵 / ∼ ) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ ))
14	13	3ad2ant2 1050	. 2 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → (𝑋 ∈ (𝐵 / ∼ ) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ ))
15	12, 14	bitrd 188	1 ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑋 ∈ 𝑊 ∧ 𝑆 ∈ 𝑍) → (𝑋 ∈ (Base‘𝑈) ↔ ∃𝑥 ∈ 𝐵 𝑋 = [𝑥] ∼ ))

Colors of variables: wff set class

This proof depends on syntax axioms: → wi 4 ↔ wb 105 ∧ w3a 1009 = wceq 1402 ∈ wcel 2209 ∃wrex 2529 Vcvv 2821 ‘cfv 5380 (class class class)co 6088 [cec 6809 / cqs 6810 Basecbs 13411 /_s cqus 13683 ~_QG cqg 14032

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-coll 4248 ax-sep 4251 ax-pow 4313 ax-pr 4348 ax-un 4580 ax-setind 4686 ax-cnex 8277 ax-resscn 8278 ax-1cn 8279 ax-1re 8280 ax-icn 8281 ax-addcl 8282 ax-addrcl 8283 ax-mulcl 8284 ax-addcom 8286 ax-addass 8288 ax-i2m1 8291 ax-0lt1 8292 ax-0id 8294 ax-rnegex 8295 ax-pre-ltirr 8298 ax-pre-lttrn 8300 ax-pre-ltadd 8302

This proof depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-csb 3148 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-nul 3521 df-pw 3690 df-sn 3715 df-pr 3716 df-tp 3717 df-op 3718 df-uni 3936 df-int 3971 df-iun 4014 df-br 4132 df-opab 4194 df-mpt 4195 df-id 4440 df-xp 4782 df-rel 4783 df-cnv 4784 df-co 4785 df-dm 4786 df-rn 4787 df-res 4788 df-ima 4789 df-iota 5340 df-fun 5382 df-fn 5383 df-f 5384 df-f1 5385 df-fo 5386 df-f1o 5387 df-fv 5388 df-ov 6091 df-oprab 6092 df-mpo 6093 df-ec 6813 df-qs 6817 df-pnf 8369 df-mnf 8370 df-ltxr 8372 df-inn 9315 df-2 9373 df-3 9374 df-ndx 13414 df-slot 13415 df-base 13417 df-plusg 13504 df-mulr 13505 df-iimas 13684 df-qus 13685 df-eqg 14035

This theorem is used by: (None)

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