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Theorem lsmpropd 18022
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 1089 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝜑)
2 simp12 1090 . . . . . . . . 9 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑡 ∈ 𝒫 𝐵)
32elpwid 4146 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑡𝐵)
4 simp2 1060 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑥𝑡)
53, 4sseldd 3588 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑥𝐵)
6 simp13 1091 . . . . . . . . 9 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑢 ∈ 𝒫 𝐵)
76elpwid 4146 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑢𝐵)
8 simp3 1061 . . . . . . . 8 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑦𝑢)
97, 8sseldd 3588 . . . . . . 7 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → 𝑦𝐵)
10 lsmpropd.p . . . . . . 7 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
111, 5, 9, 10syl12anc 1321 . . . . . 6 (((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) ∧ 𝑥𝑡𝑦𝑢) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
1211mpt2eq3dva 6679 . . . . 5 ((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) → (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦)) = (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦)))
1312rneqd 5318 . . . 4 ((𝜑𝑡 ∈ 𝒫 𝐵𝑢 ∈ 𝒫 𝐵) → ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐾)𝑦)) = ran (𝑥𝑡, 𝑦𝑢 ↦ (𝑥(+g𝐿)𝑦)))
1413mpt2eq3dva 6679 . . 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 4140 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐾))
17 mpt2eq12 6675 . . . 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 692 . . 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 4140 . . . 4 (𝜑 → 𝒫 𝐵 = 𝒫 (Base‘𝐿))
21 mpt2eq12 6675 . . . 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 692 . . 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 2663 . 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 2621 . . . 4 (Base‘𝐾) = (Base‘𝐾)
26 eqid 2621 . . . 4 (+g𝐾) = (+g𝐾)
27 eqid 2621 . . . 4 (LSSum‘𝐾) = (LSSum‘𝐾)
2825, 26, 27lsmfval 17985 . . 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 2621 . . . 4 (Base‘𝐿) = (Base‘𝐿)
32 eqid 2621 . . . 4 (+g𝐿) = (+g𝐿)
33 eqid 2621 . . . 4 (LSSum‘𝐿) = (LSSum‘𝐿)
3431, 32, 33lsmfval 17985 . . 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 2665 1 (𝜑 → (LSSum‘𝐾) = (LSSum‘𝐿))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  Vcvv 3189  𝒫 cpw 4135  ran crn 5080  cfv 5852  (class class class)co 6610  cmpt2 6612  Basecbs 15792  +gcplusg 15873  LSSumclsm 17981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-lsm 17983
This theorem is referenced by:  hlhillsm  36763
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