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Theorem lsmpropd 19639
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 4615 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑡𝐵)
4 simp2 1134 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑥𝑡)
53, 4sseldd 3983 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑥𝐵)
6 simp13 1202 . . . . . . . . 9 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑢 ∈ 𝒫 𝐵)
76elpwid 4615 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑢𝐵)
8 simp3 1135 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑦𝑢)
97, 8sseldd 3983 . . . . . . 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 7503 . . . . 5 ((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦)))
1312rneqd 5944 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦)))
1413mpoeq3dva 7503 . . 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 4623 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17 mpoeq12 7499 . . . 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 582 . . 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 4623 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐿))
21 mpoeq12 7499 . . . 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 582 . . 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 2776 . 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 2728 . . . 4 (Base‘𝐾) = (Base‘𝐾)
26 eqid 2728 . . . 4 (+g𝐾) = (+g𝐾)
27 eqid 2728 . . . 4 (LSSum‘𝐾) = (LSSum‘𝐾)
2825, 26, 27lsmfval 19600 . . 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 2728 . . . 4 (Base‘𝐿) = (Base‘𝐿)
32 eqid 2728 . . . 4 (+g𝐿) = (+g𝐿)
33 eqid 2728 . . . 4 (LSSum‘𝐿) = (LSSum‘𝐿)
3431, 32, 33lsmfval 19600 . . 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 2778 1 (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1533  wcel 2098  𝒫 cpw 4606  ran crn 5683  cfv 6553  (class class class)co 7426  cmpo 7428  Basecbs 17187  +gcplusg 17240  LSSumclsm 19596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-lsm 19598
This theorem is referenced by:  hlhillsm  41465
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