Theorem fxpgaeq 33126

Description: A fixed point 𝑋 is invariant under group action 𝐴 (Contributed by Thierry Arnoux, 18-Nov-2025.)

Hypotheses

Ref	Expression
fxpgaval.s	⊢ 𝑈 = (Base‘𝐺)
fxpgaval.a	⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
fxpgaeq.x	⊢ (𝜑 → 𝑋 ∈ (𝐶FixPts𝐴))
fxpgaeq.p	⊢ (𝜑 → 𝑃 ∈ 𝑈)

Assertion

Ref	Expression
fxpgaeq	⊢ (𝜑 → (𝑃𝐴𝑋) = 𝑋)

Proof of Theorem fxpgaeq

Dummy variables 𝑝 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7394	. . 3 ⊢ (𝑝 = 𝑃 → (𝑝𝐴𝑋) = (𝑃𝐴𝑋))
2	1	eqeq1d 2731	. 2 ⊢ (𝑝 = 𝑃 → ((𝑝𝐴𝑋) = 𝑋 ↔ (𝑃𝐴𝑋) = 𝑋))
3		fxpgaeq.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝐶FixPts𝐴))
4		fxpgaval.s	. . . . . 6 ⊢ 𝑈 = (Base‘𝐺)
5		fxpgaval.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (𝐺 GrpAct 𝐶))
6	4, 5	fxpgaval 33124	. . . . 5 ⊢ (𝜑 → (𝐶FixPts𝐴) = {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
7	3, 6	eleqtrd 2830	. . . 4 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥})
8		oveq2 7395	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑝𝐴𝑥) = (𝑝𝐴𝑋))
9		id 22	. . . . . . 7 ⊢ (𝑥 = 𝑋 → 𝑥 = 𝑋)
10	8, 9	eqeq12d 2745	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑝𝐴𝑥) = 𝑥 ↔ (𝑝𝐴𝑋) = 𝑋))
11	10	ralbidv 3156	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥 ↔ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
12	11	elrab 3659	. . . 4 ⊢ (𝑋 ∈ {𝑥 ∈ 𝐶 ∣ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑥) = 𝑥} ↔ (𝑋 ∈ 𝐶 ∧ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
13	7, 12	sylib 218	. . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐶 ∧ ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋))
14	13	simprd 495	. 2 ⊢ (𝜑 → ∀𝑝 ∈ 𝑈 (𝑝𝐴𝑋) = 𝑋)
15		fxpgaeq.p	. 2 ⊢ (𝜑 → 𝑃 ∈ 𝑈)
16	2, 14, 15	rspcdva 3589	1 ⊢ (𝜑 → (𝑃𝐴𝑋) = 𝑋)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405  ‘cfv 6511  (class class class)co 7387  Basecbs 17179   GrpAct cga 19221  FixPtscfxp 33120

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 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-map 8801  df-ga 19222  df-fxp 33121

This theorem is referenced by:  fxpsubm  33129