bdmetval - Intuitionistic Logic Explorer

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Theorem bdmetval 13862

Description: Value of the standard bounded metric. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)

**Hypothesis**
Ref	Expression
stdbdmet.1	⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ^*, < ))

**Assertion**
Ref	Expression
bdmetval	⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ))

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐶,𝑦 𝑥,𝐵,𝑦 𝑥,𝑅,𝑦 𝑥,𝑋,𝑦 Allowed substitution hints: 𝐷(𝑥,𝑦)

**Proof of Theorem bdmetval**
Step	Hyp	Ref	Expression
1		simprl 529	. 2 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐴 ∈ 𝑋)
2		simprr 531	. 2 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^*) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐵 ∈ 𝑋)
3		simpll 527	. . . 4 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝐶:(𝑋 × 𝑋)⟶ℝ^)
4	3, 1, 2	fovcdmd 6015	. . 3 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐶𝐵) ∈ ℝ^)
5		simplr 528	. . 3 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → 𝑅 ∈ ℝ^)
6		xrmincl 11266	. . 3 ⊢ (((𝐴𝐶𝐵) ∈ ℝ^* ∧ 𝑅 ∈ ℝ^) → inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ) ∈ ℝ^*)
7	4, 5, 6	syl2anc 411	. 2 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ) ∈ ℝ^*)
8		oveq12 5880	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥𝐶𝑦) = (𝐴𝐶𝐵))
9	8	preq1d 3675	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → {(𝑥𝐶𝑦), 𝑅} = {(𝐴𝐶𝐵), 𝑅})
10	9	infeq1d 7007	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → inf({(𝑥𝐶𝑦), 𝑅}, ℝ^, < ) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ))
11		stdbdmet.1	. . 3 ⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ inf({(𝑥𝐶𝑦), 𝑅}, ℝ^*, < ))
12	10, 11	ovmpoga 6000	. 2 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ) ∈ ℝ^) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ^*, < ))
13	1, 2, 7, 12	syl3anc 1238	1 ⊢ (((𝐶:(𝑋 × 𝑋)⟶ℝ^* ∧ 𝑅 ∈ ℝ^) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝐷𝐵) = inf({(𝐴𝐶𝐵), 𝑅}, ℝ^, < ))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 {cpr 3593 × cxp 4623 ⟶wf 5210 (class class class)co 5871 ∈ cmpo 5873 infcinf 6978 ℝ^*cxr 7986 < clt 7987

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4117 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 ax-cnex 7898 ax-resscn 7899 ax-1cn 7900 ax-1re 7901 ax-icn 7902 ax-addcl 7903 ax-addrcl 7904 ax-mulcl 7905 ax-mulrcl 7906 ax-addcom 7907 ax-mulcom 7908 ax-addass 7909 ax-mulass 7910 ax-distr 7911 ax-i2m1 7912 ax-0lt1 7913 ax-1rid 7914 ax-0id 7915 ax-rnegex 7916 ax-precex 7917 ax-cnre 7918 ax-pre-ltirr 7919 ax-pre-ltwlin 7920 ax-pre-lttrn 7921 ax-pre-apti 7922 ax-pre-ltadd 7923 ax-pre-mulgt0 7924 ax-pre-mulext 7925 ax-arch 7926 ax-caucvg 7927

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-tr 4101 df-id 4292 df-po 4295 df-iso 4296 df-iord 4365 df-on 4367 df-ilim 4368 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-f1 5219 df-fo 5220 df-f1o 5221 df-fv 5222 df-isom 5223 df-riota 5827 df-ov 5874 df-oprab 5875 df-mpo 5876 df-1st 6137 df-2nd 6138 df-recs 6302 df-frec 6388 df-sup 6979 df-inf 6980 df-pnf 7989 df-mnf 7990 df-xr 7991 df-ltxr 7992 df-le 7993 df-sub 8125 df-neg 8126 df-reap 8527 df-ap 8534 df-div 8625 df-inn 8915 df-2 8973 df-3 8974 df-4 8975 df-n0 9172 df-z 9249 df-uz 9524 df-rp 9649 df-xneg 9767 df-seqfrec 10440 df-exp 10514 df-cj 10843 df-re 10844 df-im 10845 df-rsqrt 10999 df-abs 11000

This theorem is referenced by: bdbl 13865

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