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Theorem lsmpropd 18534
Description: If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-2015.)
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 (𝜑𝐾 ∈ V)
lsmpropd.v2 (𝜑𝐿 ∈ V)
Assertion
Ref Expression
lsmpropd (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem lsmpropd
Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 1196 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝜑)
2 simp12 1197 . . . . . . . . 9 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑡 ∈ 𝒫 𝐵)
32elpwid 4471 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑡𝐵)
4 simp2 1130 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑥𝑡)
53, 4sseldd 3896 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑥𝐵)
6 simp13 1198 . . . . . . . . 9 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑢 ∈ 𝒫 𝐵)
76elpwid 4471 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑢𝐵)
8 simp3 1131 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑦𝑢)
97, 8sseldd 3896 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑦𝐵)
10 lsmpropd.p . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
111, 5, 9, 10syl12anc 833 . . . . . 6 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1211mpoeq3dva 7096 . . . . 5 ((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦)))
1312rneqd 5697 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦)))
1413mpoeq3dva 7096 . . 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 4464 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17 mpoeq12 7092 . . . 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 4464 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐿))
21 mpoeq12 7092 . . . 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 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 (𝜑𝐾 ∈ V)
25 eqid 2797 . . . 4 (Base‘𝐾) = (Base‘𝐾)
26 eqid 2797 . . . 4 (+g𝐾) = (+g𝐾)
27 eqid 2797 . . . 4 (LSSum‘𝐾) = (LSSum‘𝐾)
2825, 26, 27lsmfval 18497 . . 3 (𝐾 ∈ V → (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 (𝜑𝐿 ∈ V)
31 eqid 2797 . . . 4 (Base‘𝐿) = (Base‘𝐿)
32 eqid 2797 . . . 4 (+g𝐿) = (+g𝐿)
33 eqid 2797 . . . 4 (LSSum‘𝐿) = (LSSum‘𝐿)
3431, 32, 33lsmfval 18497 . . 3 (𝐿 ∈ V → (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 396  w3a 1080   = wceq 1525  wcel 2083  Vcvv 3440  𝒫 cpw 4459  ran crn 5451  cfv 6232  (class class class)co 7023  cmpo 7025  Basecbs 16316  +gcplusg 16398  LSSumclsm 18493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-1st 7552  df-2nd 7553  df-lsm 18495
This theorem is referenced by:  hlhillsm  38644
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