Theorem lsppropd 14567

Description: If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) (Revised by AV, 24-Apr-2024.)

Hypotheses

Ref	Expression
lsspropd.b1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
lsspropd.b2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
lsspropd.w	⊢ (𝜑 → 𝐵 ⊆ 𝑊)
lsspropd.p	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
lsspropd.s1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)
lsspropd.s2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
lsspropd.p1	⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))
lsspropd.p2	⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
lsppropd.v1	⊢ (𝜑 → 𝐾 ∈ 𝑋)
lsppropd.v2	⊢ (𝜑 → 𝐿 ∈ 𝑌)

Assertion

Ref	Expression
lsppropd	⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝜑,𝑥,𝑦   𝑥,𝑊,𝑦   𝑥,𝐿,𝑦   𝑥,𝑃,𝑦

Allowed substitution hints:   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦)

Proof of Theorem lsppropd

Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lsspropd.b1	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		lsspropd.b2	. . . . 5 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3	1, 2	eqtr3d 2267	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4	3	pweqd 3673	. . 3 ⊢ (𝜑 → 𝒫 (Base‘𝐾) = 𝒫 (Base‘𝐿))
5		lsspropd.w	. . . . . 6 ⊢ (𝜑 → 𝐵 ⊆ 𝑊)
6		lsspropd.p	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7		lsspropd.s1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊)
8		lsspropd.s2	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦))
9		lsspropd.p1	. . . . . 6 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾)))
10		lsspropd.p2	. . . . . 6 ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿)))
11		lsppropd.v1	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ 𝑋)
12		lsppropd.v2	. . . . . 6 ⊢ (𝜑 → 𝐿 ∈ 𝑌)
13	1, 2, 5, 6, 7, 8, 9, 10, 11, 12	lsspropdg 14566	. . . . 5 ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))
14	13	rabeqdv 2806	. . . 4 ⊢ (𝜑 → {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡})
15	14	inteqd 3953	. . 3 ⊢ (𝜑 → ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡})
16	4, 15	mpteq12dv 4191	. 2 ⊢ (𝜑 → (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}))
17		eqid 2232	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2232	. . . 4 ⊢ (LSubSp‘𝐾) = (LSubSp‘𝐾)
19		eqid 2232	. . . 4 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
20	17, 18, 19	lspfval 14523	. . 3 ⊢ (𝐾 ∈ 𝑋 → (LSpan‘𝐾) = (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡}))
21	11, 20	syl 14	. 2 ⊢ (𝜑 → (LSpan‘𝐾) = (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡}))
22		eqid 2232	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
23		eqid 2232	. . . 4 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿)
24		eqid 2232	. . . 4 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿)
25	22, 23, 24	lspfval 14523	. . 3 ⊢ (𝐿 ∈ 𝑌 → (LSpan‘𝐿) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}))
26	12, 25	syl 14	. 2 ⊢ (𝜑 → (LSpan‘𝐿) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}))
27	16, 21, 26	3eqtr4d 2275	1 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2203  {crab 2524   ⊆ wss 3210  𝒫 cpw 3668  ∩ cint 3948   ↦ cmpt 4170  ‘cfv 5351  (class class class)co 6049  Basecbs 13201  +_gcplusg 13279  Scalarcsca 13282   ·_𝑠 cvsca 13283  LSubSpclss 14487  LSpanclspn 14521

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-cnex 8214  ax-resscn 8215  ax-1re 8217  ax-addrcl 8220

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-id 4413  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-ov 6052  df-inn 9234  df-ndx 13204  df-slot 13205  df-base 13207  df-lssm 14488  df-lsp 14522

This theorem is referenced by:  lidlrsppropdg  14630