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Theorem lsmpropd 18816
 Description: If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.) (Revised by AV, 25-Apr-2024.)
Hypotheses
Ref Expression
lsmpropd.b1 (𝜑𝐵 = (Base‘𝐾))
lsmpropd.b2 (𝜑𝐵 = (Base‘𝐿))
lsmpropd.p ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
lsmpropd.v1 (𝜑𝐾𝑉)
lsmpropd.v2 (𝜑𝐿𝑊)
Assertion
Ref Expression
lsmpropd (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem lsmpropd
Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 1200 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝜑)
2 simp12 1201 . . . . . . . . 9 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑡 ∈ 𝒫 𝐵)
32elpwid 4511 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑡𝐵)
4 simp2 1134 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑥𝑡)
53, 4sseldd 3918 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑥𝐵)
6 simp13 1202 . . . . . . . . 9 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑢 ∈ 𝒫 𝐵)
76elpwid 4511 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑢𝐵)
8 simp3 1135 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑦𝑢)
97, 8sseldd 3918 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑦𝐵)
10 lsmpropd.p . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
111, 5, 9, 10syl12anc 835 . . . . . 6 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1211mpoeq3dva 7220 . . . . 5 ((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦)))
1312rneqd 5778 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦)))
1413mpoeq3dva 7220 . . 3 (𝜑 → (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦))) = (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦))))
15 lsmpropd.b1 . . . . 5 (𝜑𝐵 = (Base‘𝐾))
1615pweqd 4519 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17 mpoeq12 7216 . . . 4 ((𝒫 𝐵 = 𝒫 (Base‘𝐾) ∧ 𝒫 𝐵 = 𝒫 (Base‘𝐾)) → (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐾), 𝑢 ∈ 𝒫 (Base‘𝐾) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦))))
1816, 16, 17syl2anc 587 . . 3 (𝜑 → (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐾), 𝑢 ∈ 𝒫 (Base‘𝐾) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦))))
19 lsmpropd.b2 . . . . 5 (𝜑𝐵 = (Base‘𝐿))
2019pweqd 4519 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐿))
21 mpoeq12 7216 . . . 4 ((𝒫 𝐵 = 𝒫 (Base‘𝐿) ∧ 𝒫 𝐵 = 𝒫 (Base‘𝐿)) → (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐿), 𝑢 ∈ 𝒫 (Base‘𝐿) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦))))
2220, 20, 21syl2anc 587 . . 3 (𝜑 → (𝑡 ∈ 𝒫 𝐵, 𝑢 ∈ 𝒫 𝐵 ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐿), 𝑢 ∈ 𝒫 (Base‘𝐿) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦))))
2314, 18, 223eqtr3d 2841 . 2 (𝜑 → (𝑡 ∈ 𝒫 (Base‘𝐾), 𝑢 ∈ 𝒫 (Base‘𝐾) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦))) = (𝑡 ∈ 𝒫 (Base‘𝐿), 𝑢 ∈ 𝒫 (Base‘𝐿) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦))))
24 lsmpropd.v1 . . 3 (𝜑𝐾𝑉)
25 eqid 2798 . . . 4 (Base‘𝐾) = (Base‘𝐾)
26 eqid 2798 . . . 4 (+g𝐾) = (+g𝐾)
27 eqid 2798 . . . 4 (LSSum‘𝐾) = (LSSum‘𝐾)
2825, 26, 27lsmfval 18776 . . 3 (𝐾𝑉 → (LSSum‘𝐾) = (𝑡 ∈ 𝒫 (Base‘𝐾), 𝑢 ∈ 𝒫 (Base‘𝐾) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦))))
2924, 28syl 17 . 2 (𝜑 → (LSSum‘𝐾) = (𝑡 ∈ 𝒫 (Base‘𝐾), 𝑢 ∈ 𝒫 (Base‘𝐾) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦))))
30 lsmpropd.v2 . . 3 (𝜑𝐿𝑊)
31 eqid 2798 . . . 4 (Base‘𝐿) = (Base‘𝐿)
32 eqid 2798 . . . 4 (+g𝐿) = (+g𝐿)
33 eqid 2798 . . . 4 (LSSum‘𝐿) = (LSSum‘𝐿)
3431, 32, 33lsmfval 18776 . . 3 (𝐿𝑊 → (LSSum‘𝐿) = (𝑡 ∈ 𝒫 (Base‘𝐿), 𝑢 ∈ 𝒫 (Base‘𝐿) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦))))
3530, 34syl 17 . 2 (𝜑 → (LSSum‘𝐿) = (𝑡 ∈ 𝒫 (Base‘𝐿), 𝑢 ∈ 𝒫 (Base‘𝐿) ↦ ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦))))
3623, 29, 353eqtr4d 2843 1 (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  𝒫 cpw 4500  ran crn 5524  ‘cfv 6332  (class class class)co 7145   ∈ cmpo 7147  Basecbs 16495  +gcplusg 16577  LSSumclsm 18772 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5158  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6291  df-fun 6334  df-fn 6335  df-f 6336  df-f1 6337  df-fo 6338  df-f1o 6339  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7684  df-2nd 7685  df-lsm 18774 This theorem is referenced by:  hlhillsm  39403
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