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Theorem lsmpropd 19695
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 1204 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝜑)
2 simp12 1205 . . . . . . . . 9 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑡 ∈ 𝒫 𝐵)
32elpwid 4609 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑡𝐵)
4 simp2 1138 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑥𝑡)
53, 4sseldd 3984 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑥𝐵)
6 simp13 1206 . . . . . . . . 9 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑢 ∈ 𝒫 𝐵)
76elpwid 4609 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑢𝐵)
8 simp3 1139 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑦𝑢)
97, 8sseldd 3984 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑦𝐵)
10 lsmpropd.p . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
111, 5, 9, 10syl12anc 837 . . . . . 6 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1211mpoeq3dva 7510 . . . . 5 ((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦)))
1312rneqd 5949 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦)))
1413mpoeq3dva 7510 . . 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 4617 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17 mpoeq12 7506 . . . 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 584 . . 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 4617 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐿))
21 mpoeq12 7506 . . . 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 584 . . 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 2785 . 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 2737 . . . 4 (Base‘𝐾) = (Base‘𝐾)
26 eqid 2737 . . . 4 (+g𝐾) = (+g𝐾)
27 eqid 2737 . . . 4 (LSSum‘𝐾) = (LSSum‘𝐾)
2825, 26, 27lsmfval 19656 . . 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 2737 . . . 4 (Base‘𝐿) = (Base‘𝐿)
32 eqid 2737 . . . 4 (+g𝐿) = (+g𝐿)
33 eqid 2737 . . . 4 (LSSum‘𝐿) = (LSSum‘𝐿)
3431, 32, 33lsmfval 19656 . . 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 2787 1 (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  𝒫 cpw 4600  ran crn 5686  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  +gcplusg 17297  LSSumclsm 19652
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-lsm 19654
This theorem is referenced by:  hlhillsm  41962
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