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Theorem isismt 28356
Description: Property of being an isometry. Compare with isismty 37279. (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 3490 . . . 4 (𝐺 ∈ 𝑉 β†’ 𝐺 ∈ V)
2 elex 3490 . . . 4 (𝐻 ∈ π‘Š β†’ 𝐻 ∈ V)
3 fveq2 6900 . . . . . . . . 9 (𝑔 = 𝐺 β†’ (Baseβ€˜π‘”) = (Baseβ€˜πΊ))
4 isismt.b . . . . . . . . 9 𝐡 = (Baseβ€˜πΊ)
53, 4eqtr4di 2785 . . . . . . . 8 (𝑔 = 𝐺 β†’ (Baseβ€˜π‘”) = 𝐡)
65f1oeq2d 6838 . . . . . . 7 (𝑔 = 𝐺 β†’ (𝑓:(Baseβ€˜π‘”)–1-1-ontoβ†’(Baseβ€˜β„Ž) ↔ 𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜β„Ž)))
7 fveq2 6900 . . . . . . . . . . . 12 (𝑔 = 𝐺 β†’ (distβ€˜π‘”) = (distβ€˜πΊ))
8 isismt.d . . . . . . . . . . . 12 𝐷 = (distβ€˜πΊ)
97, 8eqtr4di 2785 . . . . . . . . . . 11 (𝑔 = 𝐺 β†’ (distβ€˜π‘”) = 𝐷)
109oveqd 7441 . . . . . . . . . 10 (𝑔 = 𝐺 β†’ (π‘Ž(distβ€˜π‘”)𝑏) = (π‘Žπ·π‘))
1110eqeq2d 2738 . . . . . . . . 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 3338 . . . . . . . 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 3338 . . . . . . 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 630 . . . . . 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 2796 . . . . 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 6900 . . . . . . . . 9 (β„Ž = 𝐻 β†’ (Baseβ€˜β„Ž) = (Baseβ€˜π»))
17 isismt.p . . . . . . . . 9 𝑃 = (Baseβ€˜π»)
1816, 17eqtr4di 2785 . . . . . . . 8 (β„Ž = 𝐻 β†’ (Baseβ€˜β„Ž) = 𝑃)
1918f1oeq3d 6839 . . . . . . 7 (β„Ž = 𝐻 β†’ (𝑓:𝐡–1-1-ontoβ†’(Baseβ€˜β„Ž) ↔ 𝑓:𝐡–1-1-onto→𝑃))
20 fveq2 6900 . . . . . . . . . . 11 (β„Ž = 𝐻 β†’ (distβ€˜β„Ž) = (distβ€˜π»))
21 isismt.m . . . . . . . . . . 11 βˆ’ = (distβ€˜π»)
2220, 21eqtr4di 2785 . . . . . . . . . 10 (β„Ž = 𝐻 β†’ (distβ€˜β„Ž) = βˆ’ )
2322oveqd 7441 . . . . . . . . 9 (β„Ž = 𝐻 β†’ ((π‘“β€˜π‘Ž)(distβ€˜β„Ž)(π‘“β€˜π‘)) = ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)))
2423eqeq1d 2729 . . . . . . . 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 3214 . . . . . . 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 630 . . . . . 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 2796 . . . . 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 28355 . . . . 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 7457 . . . . . 6 (𝑃 ↑m 𝐡) ∈ V
30 f1of 6842 . . . . . . . . 9 (𝑓:𝐡–1-1-onto→𝑃 β†’ 𝑓:π΅βŸΆπ‘ƒ)
3117fvexi 6914 . . . . . . . . . 10 𝑃 ∈ V
324fvexi 6914 . . . . . . . . . 10 𝐡 ∈ V
3331, 32elmap 8894 . . . . . . . . 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 479 . . . . . . 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 4065 . . . . . 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 5324 . . . . 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 7585 . . . 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 594 . . 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 2814 . 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 6842 . . . . 5 (𝐹:𝐡–1-1-onto→𝑃 β†’ 𝐹:π΅βŸΆπ‘ƒ)
42 fex 7242 . . . . 5 ((𝐹:π΅βŸΆπ‘ƒ ∧ 𝐡 ∈ V) β†’ 𝐹 ∈ V)
4341, 32, 42sylancl 584 . . . 4 (𝐹:𝐡–1-1-onto→𝑃 β†’ 𝐹 ∈ V)
4443adantr 479 . . 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 6830 . . . 4 (𝑓 = 𝐹 β†’ (𝑓:𝐡–1-1-onto→𝑃 ↔ 𝐹:𝐡–1-1-onto→𝑃))
46 fveq1 6899 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘Ž) = (πΉβ€˜π‘Ž))
47 fveq1 6899 . . . . . . 7 (𝑓 = 𝐹 β†’ (π‘“β€˜π‘) = (πΉβ€˜π‘))
4846, 47oveq12d 7442 . . . . . 6 (𝑓 = 𝐹 β†’ ((π‘“β€˜π‘Ž) βˆ’ (π‘“β€˜π‘)) = ((πΉβ€˜π‘Ž) βˆ’ (πΉβ€˜π‘)))
4948eqeq1d 2729 . . . . 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 3214 . . . 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 630 . . 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 3675 . 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 286 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 394   = wceq 1533   ∈ wcel 2098  {cab 2704  βˆ€wral 3057  Vcvv 3471  βŸΆwf 6547  β€“1-1-ontoβ†’wf1o 6550  β€˜cfv 6551  (class class class)co 7424   ↑m cmap 8849  Basecbs 17185  distcds 17247  Ismtcismt 28354
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 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
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 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8851  df-ismt 28355
This theorem is referenced by:  ismot  28357
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