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Theorem gtiso 30349
Description: Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)
Assertion
Ref Expression
gtiso ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))

Proof of Theorem gtiso
StepHypRef Expression
1 eqid 2826 . . . . 5 ((𝐴 × 𝐴) ∖ < ) = ((𝐴 × 𝐴) ∖ < )
2 eqid 2826 . . . . 5 ((𝐵 × 𝐵) ∖ < ) = ((𝐵 × 𝐵) ∖ < )
31, 2isocnv3 7077 . . . 4 (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵))
43a1i 11 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
5 df-le 10670 . . . . . . . . . 10 ≤ = ((ℝ* × ℝ*) ∖ < )
65cnveqi 5744 . . . . . . . . 9 ≤ = ((ℝ* × ℝ*) ∖ < )
7 cnvdif 6000 . . . . . . . . 9 ((ℝ* × ℝ*) ∖ < ) = ((ℝ* × ℝ*) ∖ < )
8 cnvxp 6012 . . . . . . . . . 10 (ℝ* × ℝ*) = (ℝ* × ℝ*)
9 ltrel 10692 . . . . . . . . . . 11 Rel <
10 dfrel2 6044 . . . . . . . . . . 11 (Rel < ↔ < = < )
119, 10mpbi 231 . . . . . . . . . 10 < = <
128, 11difeq12i 4101 . . . . . . . . 9 ((ℝ* × ℝ*) ∖ < ) = ((ℝ* × ℝ*) ∖ < )
136, 7, 123eqtri 2853 . . . . . . . 8 ≤ = ((ℝ* × ℝ*) ∖ < )
1413ineq1i 4189 . . . . . . 7 ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∖ < ) ∩ (𝐴 × 𝐴))
15 indif1 4252 . . . . . . 7 (((ℝ* × ℝ*) ∖ < ) ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < )
1614, 15eqtri 2849 . . . . . 6 ( ≤ ∩ (𝐴 × 𝐴)) = (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < )
17 xpss12 5569 . . . . . . . . 9 ((𝐴 ⊆ ℝ*𝐴 ⊆ ℝ*) → (𝐴 × 𝐴) ⊆ (ℝ* × ℝ*))
1817anidms 567 . . . . . . . 8 (𝐴 ⊆ ℝ* → (𝐴 × 𝐴) ⊆ (ℝ* × ℝ*))
19 sseqin2 4196 . . . . . . . 8 ((𝐴 × 𝐴) ⊆ (ℝ* × ℝ*) ↔ ((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
2018, 19sylib 219 . . . . . . 7 (𝐴 ⊆ ℝ* → ((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) = (𝐴 × 𝐴))
2120difeq1d 4102 . . . . . 6 (𝐴 ⊆ ℝ* → (((ℝ* × ℝ*) ∩ (𝐴 × 𝐴)) ∖ < ) = ((𝐴 × 𝐴) ∖ < ))
2216, 21syl5req 2874 . . . . 5 (𝐴 ⊆ ℝ* → ((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
2322adantr 481 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)))
24 isoeq2 7063 . . . 4 (((𝐴 × 𝐴) ∖ < ) = ( ≤ ∩ (𝐴 × 𝐴)) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
2523, 24syl 17 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom ((𝐴 × 𝐴) ∖ < ), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵)))
265ineq1i 4189 . . . . . . 7 ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∖ < ) ∩ (𝐵 × 𝐵))
27 indif1 4252 . . . . . . 7 (((ℝ* × ℝ*) ∖ < ) ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < )
2826, 27eqtri 2849 . . . . . 6 ( ≤ ∩ (𝐵 × 𝐵)) = (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < )
29 xpss12 5569 . . . . . . . . 9 ((𝐵 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐵 × 𝐵) ⊆ (ℝ* × ℝ*))
3029anidms 567 . . . . . . . 8 (𝐵 ⊆ ℝ* → (𝐵 × 𝐵) ⊆ (ℝ* × ℝ*))
31 sseqin2 4196 . . . . . . . 8 ((𝐵 × 𝐵) ⊆ (ℝ* × ℝ*) ↔ ((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
3230, 31sylib 219 . . . . . . 7 (𝐵 ⊆ ℝ* → ((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) = (𝐵 × 𝐵))
3332difeq1d 4102 . . . . . 6 (𝐵 ⊆ ℝ* → (((ℝ* × ℝ*) ∩ (𝐵 × 𝐵)) ∖ < ) = ((𝐵 × 𝐵) ∖ < ))
3428, 33syl5req 2874 . . . . 5 (𝐵 ⊆ ℝ* → ((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
3534adantl 482 . . . 4 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → ((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)))
36 isoeq3 7064 . . . 4 (((𝐵 × 𝐵) ∖ < ) = ( ≤ ∩ (𝐵 × 𝐵)) → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
3735, 36syl 17 . . 3 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ((𝐵 × 𝐵) ∖ < )(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
384, 25, 373bitrd 306 . 2 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵)))
39 isocnv2 7076 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))
40 isores2 7078 . . . 4 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
41 isores1 7079 . . . 4 (𝐹 Isom ≤ , ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
4240, 41bitri 276 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
43 lerel 10694 . . . . 5 Rel ≤
44 dfrel2 6044 . . . . 5 (Rel ≤ ↔ ≤ = ≤ )
4543, 44mpbi 231 . . . 4 ≤ = ≤
46 isoeq2 7063 . . . 4 ( ≤ = ≤ → (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
4745, 46ax-mp 5 . . 3 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵))
4839, 42, 473bitr3ri 303 . 2 (𝐹 Isom ≤ , ≤ (𝐴, 𝐵) ↔ 𝐹 Isom ( ≤ ∩ (𝐴 × 𝐴)), ( ≤ ∩ (𝐵 × 𝐵))(𝐴, 𝐵))
4938, 48syl6bbr 290 1 ((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  cdif 3937  cin 3939  wss 3940   × cxp 5552  ccnv 5553  Rel wrel 5559   Isom wiso 6353  *cxr 10663   < clt 10664  cle 10665
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-isom 6361  df-xr 10668  df-ltxr 10669  df-le 10670
This theorem is referenced by:  xrge0iifhmeo  31065
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