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Theorem isismt 27518
Description: Property of being an isometry. Compare with isismty 36306. (Contributed by Thierry Arnoux, 13-Dec-2019.)
Hypotheses
Ref Expression
isismt.b 𝐡 = (Baseβ€˜πΊ)
isismt.p 𝑃 = (Baseβ€˜π»)
isismt.d 𝐷 = (distβ€˜πΊ)
isismt.m βˆ’ = (distβ€˜π»)
Assertion
Ref Expression
isismt ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ π‘Š) β†’ (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Žπ·π‘))))
Distinct variable groups:   𝐡,π‘Ž,𝑏   𝐹,π‘Ž,𝑏   𝐺,π‘Ž,𝑏   𝐻,π‘Ž,𝑏
Allowed substitution hints:   𝐷(π‘Ž,𝑏)   𝑃(π‘Ž,𝑏)   βˆ’ (π‘Ž,𝑏)   𝑉(π‘Ž,𝑏)   π‘Š(π‘Ž,𝑏)

Proof of Theorem isismt
Dummy variables 𝑓 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3462 . . . 4 (𝐺 ∈ 𝑉 β†’ 𝐺 ∈ V)
2 elex 3462 . . . 4 (𝐻 ∈ π‘Š β†’ 𝐻 ∈ V)
3 fveq2 6843 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (Baseβ€˜π‘”) = (Baseβ€˜πΊ))
4 isismt.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΊ)
53, 4eqtr4di 2791 . . . . . . . 8 (𝑔 = 𝐺 β†’ (Baseβ€˜π‘”) = 𝐡)
65f1oeq2d 6781 . . . . . . 7 (𝑔 = 𝐺 β†’ (𝑓:(Baseβ€˜π‘”)–1-1-ontoβ†’(Baseβ€˜β„Ž) ↔ 𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜β„Ž)))
7 fveq2 6843 . . . . . . . . . . . 12 (𝑔 = 𝐺 β†’ (distβ€˜π‘”) = (distβ€˜πΊ))
8 isismt.d . . . . . . . . . . . 12 𝐷 = (distβ€˜πΊ)
97, 8eqtr4di 2791 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ (distβ€˜π‘”) = 𝐷)
109oveqd 7375 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (π‘Ž(distβ€˜π‘”)𝑏) = (π‘Žπ·π‘))
1110eqeq2d 2744 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Ž(distβ€˜π‘”)𝑏) ↔ ((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Žπ·π‘)))
125, 11raleqbidv 3318 . . . . . . . 8 (𝑔 = 𝐺 β†’ (βˆ€π‘ ∈ (Baseβ€˜π‘”)((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Ž(distβ€˜π‘”)𝑏) ↔ βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Žπ·π‘)))
135, 12raleqbidv 3318 . . . . . . 7 (𝑔 = 𝐺 β†’ (βˆ€π‘Ž ∈ (Baseβ€˜π‘”)βˆ€π‘ ∈ (Baseβ€˜π‘”)((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Ž(distβ€˜π‘”)𝑏) ↔ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Žπ·π‘)))
146, 13anbi12d 632 . . . . . 6 (𝑔 = 𝐺 β†’ ((𝑓:(Baseβ€˜π‘”)–1-1-ontoβ†’(Baseβ€˜β„Ž) ∧ βˆ€π‘Ž ∈ (Baseβ€˜π‘”)βˆ€π‘ ∈ (Baseβ€˜π‘”)((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Ž(distβ€˜π‘”)𝑏)) ↔ (𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜β„Ž) ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Žπ·π‘))))
1514abbidv 2802 . . . . 5 (𝑔 = 𝐺 β†’ {𝑓 ∣ (𝑓:(Baseβ€˜π‘”)–1-1-ontoβ†’(Baseβ€˜β„Ž) ∧ βˆ€π‘Ž ∈ (Baseβ€˜π‘”)βˆ€π‘ ∈ (Baseβ€˜π‘”)((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Ž(distβ€˜π‘”)𝑏))} = {𝑓 ∣ (𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜β„Ž) ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Žπ·π‘))})
16 fveq2 6843 . . . . . . . . 9 (β„Ž = 𝐻 β†’ (Baseβ€˜β„Ž) = (Baseβ€˜π»))
17 isismt.p . . . . . . . . 9 𝑃 = (Baseβ€˜π»)
1816, 17eqtr4di 2791 . . . . . . . 8 (β„Ž = 𝐻 β†’ (Baseβ€˜β„Ž) = 𝑃)
1918f1oeq3d 6782 . . . . . . 7 (β„Ž = 𝐻 β†’ (𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜β„Ž) ↔ 𝑓:𝐡–1-1-onto→𝑃))
20 fveq2 6843 . . . . . . . . . . 11 (β„Ž = 𝐻 β†’ (distβ€˜β„Ž) = (distβ€˜π»))
21 isismt.m . . . . . . . . . . 11 βˆ’ = (distβ€˜π»)
2220, 21eqtr4di 2791 . . . . . . . . . 10 (β„Ž = 𝐻 β†’ (distβ€˜β„Ž) = βˆ’ )
2322oveqd 7375 . . . . . . . . 9 (β„Ž = 𝐻 β†’ ((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)))
2423eqeq1d 2735 . . . . . . . 8 (β„Ž = 𝐻 β†’ (((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Žπ·π‘) ↔ ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘)))
25242ralbidv 3209 . . . . . . 7 (β„Ž = 𝐻 β†’ (βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Žπ·π‘) ↔ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘)))
2619, 25anbi12d 632 . . . . . 6 (β„Ž = 𝐻 β†’ ((𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜β„Ž) ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Žπ·π‘)) ↔ (𝑓:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘))))
2726abbidv 2802 . . . . 5 (β„Ž = 𝐻 β†’ {𝑓 ∣ (𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜β„Ž) ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Žπ·π‘))} = {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘))})
28 df-ismt 27517 . . . . 5 Ismt = (𝑔 ∈ V, β„Ž ∈ V ↦ {𝑓 ∣ (𝑓:(Baseβ€˜π‘”)–1-1-ontoβ†’(Baseβ€˜β„Ž) ∧ βˆ€π‘Ž ∈ (Baseβ€˜π‘”)βˆ€π‘ ∈ (Baseβ€˜π‘”)((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = (π‘Ž(distβ€˜π‘”)𝑏))})
29 ovex 7391 . . . . . 6 (𝑃 ↑m 𝐡) ∈ V
30 f1of 6785 . . . . . . . . 9 (𝑓:𝐡–1-1-onto→𝑃 β†’ 𝑓:π΅βŸΆπ‘ƒ)
3117fvexi 6857 . . . . . . . . . 10 𝑃 ∈ V
324fvexi 6857 . . . . . . . . . 10 𝐡 ∈ V
3331, 32elmap 8812 . . . . . . . . 9 (𝑓 ∈ (𝑃 ↑m 𝐡) ↔ 𝑓:π΅βŸΆπ‘ƒ)
3430, 33sylibr 233 . . . . . . . 8 (𝑓:𝐡–1-1-onto→𝑃 β†’ 𝑓 ∈ (𝑃 ↑m 𝐡))
3534adantr 482 . . . . . . 7 ((𝑓:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘)) β†’ 𝑓 ∈ (𝑃 ↑m 𝐡))
3635abssi 4028 . . . . . 6 {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘))} βŠ† (𝑃 ↑m 𝐡)
3729, 36ssexi 5280 . . . . 5 {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘))} ∈ V
3815, 27, 28, 37ovmpo 7516 . . . 4 ((𝐺 ∈ V ∧ 𝐻 ∈ V) β†’ (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘))})
391, 2, 38syl2an 597 . . 3 ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ π‘Š) β†’ (𝐺Ismt𝐻) = {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘))})
4039eleq2d 2820 . 2 ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ π‘Š) β†’ (𝐹 ∈ (𝐺Ismt𝐻) ↔ 𝐹 ∈ {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘))}))
41 f1of 6785 . . . . 5 (𝐹:𝐡–1-1-onto→𝑃 β†’ 𝐹:π΅βŸΆπ‘ƒ)
42 fex 7177 . . . . 5 ((𝐹:π΅βŸΆπ‘ƒ ∧ 𝐡 ∈ V) β†’ 𝐹 ∈ V)
4341, 32, 42sylancl 587 . . . 4 (𝐹:𝐡–1-1-onto→𝑃 β†’ 𝐹 ∈ V)
4443adantr 482 . . 3 ((𝐹:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Žπ·π‘)) β†’ 𝐹 ∈ V)
45 f1oeq1 6773 . . . 4 (𝑓 = 𝐹 β†’ (𝑓:𝐡–1-1-onto→𝑃 ↔ 𝐹:𝐡–1-1-onto→𝑃))
46 fveq1 6842 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘Ž) = (πΉβ€˜π‘Ž))
47 fveq1 6842 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘) = (πΉβ€˜π‘))
4846, 47oveq12d 7376 . . . . . 6 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)))
4948eqeq1d 2735 . . . . 5 (𝑓 = 𝐹 β†’ (((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘) ↔ ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Žπ·π‘)))
50492ralbidv 3209 . . . 4 (𝑓 = 𝐹 β†’ (βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘) ↔ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Žπ·π‘)))
5145, 50anbi12d 632 . . 3 (𝑓 = 𝐹 β†’ ((𝑓:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘)) ↔ (𝐹:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Žπ·π‘))))
5244, 51elab3 3639 . 2 (𝐹 ∈ {𝑓 ∣ (𝑓:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = (π‘Žπ·π‘))} ↔ (𝐹:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Žπ·π‘)))
5340, 52bitrdi 287 1 ((𝐺 ∈ 𝑉 ∧ 𝐻 ∈ π‘Š) β†’ (𝐹 ∈ (𝐺Ismt𝐻) ↔ (𝐹:𝐡–1-1-onto→𝑃 ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ 𝐡 ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)) = (π‘Žπ·π‘))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  {cab 2710  βˆ€wral 3061  Vcvv 3444  βŸΆwf 6493  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  (class class class)co 7358   ↑m cmap 8768  Basecbs 17088  distcds 17147  Ismtcismt 27516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8770  df-ismt 27517
This theorem is referenced by:  ismot  27519
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