Theorem lsppropd 13678

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 2222	. . . 4 ⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿))
4	3	pweqd 3592	. . 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 13677	. . . . 5 ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))
14	13	rabeqdv 2743	. . . 4 ⊢ (𝜑 → {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡})
15	14	inteqd 3861	. . 3 ⊢ (𝜑 → ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡} = ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡})
16	4, 15	mpteq12dv 4097	. 2 ⊢ (𝜑 → (𝑠 ∈ 𝒫 (Base‘𝐾) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐾) ∣ 𝑠 ⊆ 𝑡}) = (𝑠 ∈ 𝒫 (Base‘𝐿) ↦ ∩ {𝑡 ∈ (LSubSp‘𝐿) ∣ 𝑠 ⊆ 𝑡}))
17		eqid 2187	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2187	. . . 4 ⊢ (LSubSp‘𝐾) = (LSubSp‘𝐾)
19		eqid 2187	. . . 4 ⊢ (LSpan‘𝐾) = (LSpan‘𝐾)
20	17, 18, 19	lspfval 13634	. . 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 2187	. . . 4 ⊢ (Base‘𝐿) = (Base‘𝐿)
23		eqid 2187	. . . 4 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿)
24		eqid 2187	. . . 4 ⊢ (LSpan‘𝐿) = (LSpan‘𝐿)
25	22, 23, 24	lspfval 13634	. . 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 2230	1 ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1363   ∈ wcel 2158  {crab 2469   ⊆ wss 3141  𝒫 cpw 3587  ∩ cint 3856   ↦ cmpt 4076  ‘cfv 5228  (class class class)co 5888  Basecbs 12476  +_gcplusg 12551  Scalarcsca 12554   ·_𝑠 cvsca 12555  LSubSpclss 13598  LSpanclspn 13632

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-cnex 7916  ax-resscn 7917  ax-1re 7919  ax-addrcl 7922

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-inn 8934  df-ndx 12479  df-slot 12480  df-base 12482  df-lssm 13599  df-lsp 13633

This theorem is referenced by:  lidlrsppropdg  13741