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Theorem prdsdsval 17189

Description: Value of the metric in a structure product. (Contributed by Mario Carneiro, 20-Aug-2015.)

**Hypotheses**
Ref	Expression
prdsbasmpt.y	⊢ 𝑌 = (𝑆X_s𝑅)
prdsbasmpt.b	⊢ 𝐵 = (Base‘𝑌)
prdsbasmpt.s	⊢ (𝜑 → 𝑆 ∈ 𝑉)
prdsbasmpt.i	⊢ (𝜑 → 𝐼 ∈ 𝑊)
prdsbasmpt.r	⊢ (𝜑 → 𝑅 Fn 𝐼)
prdsplusgval.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
prdsplusgval.g	⊢ (𝜑 → 𝐺 ∈ 𝐵)
prdsdsval.d	⊢ 𝐷 = (dist‘𝑌)

**Assertion**
Ref	Expression
prdsdsval	⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ))

Distinct variable groups: 𝑥,𝐵 𝑥,𝐹 𝑥,𝐺 𝜑,𝑥 𝑥,𝐼 𝑥,𝑉 𝑥,𝑅 𝑥,𝑆 𝑥,𝑊 𝑥,𝑌 Allowed substitution hint: 𝐷(𝑥)

Proof of Theorem prdsdsval Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsbasmpt.y	. . 3 ⊢ 𝑌 = (𝑆X_s𝑅)
2		prdsbasmpt.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
3		prdsbasmpt.r	. . . 4 ⊢ (𝜑 → 𝑅 Fn 𝐼)
4		prdsbasmpt.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑊)
5		fnex 7093	. . . 4 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑊) → 𝑅 ∈ V)
6	3, 4, 5	syl2anc 584	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
7		prdsbasmpt.b	. . 3 ⊢ 𝐵 = (Base‘𝑌)
8		fndm 6536	. . . 4 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
9	3, 8	syl 17	. . 3 ⊢ (𝜑 → dom 𝑅 = 𝐼)
10		prdsdsval.d	. . 3 ⊢ 𝐷 = (dist‘𝑌)
11	1, 2, 6, 7, 9, 10	prdsds 17175	. 2 ⊢ (𝜑 → 𝐷 = (𝑓 ∈ 𝐵, 𝑔 ∈ 𝐵 ↦ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^*, < )))
12		fveq1 6773	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
13		fveq1 6773	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑔‘𝑥) = (𝐺‘𝑥))
14	12, 13	oveqan12d 7294	. . . . . . 7 ⊢ ((𝑓 = 𝐹 ∧ 𝑔 = 𝐺) → ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)) = ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥)))
15	14	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥)) = ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥)))
16	15	mpteq2dv 5176	. . . . 5 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))))
17	16	rneqd 5847	. . . 4 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) = ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))))
18	17	uneq1d 4096	. . 3 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → (ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}) = (ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}))
19	18	supeq1d 9205	. 2 ⊢ ((𝜑 ∧ (𝑓 = 𝐹 ∧ 𝑔 = 𝐺)) → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥)(dist‘(𝑅‘𝑥))(𝑔‘𝑥))) ∪ {0}), ℝ^, < ) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^, < ))
20		prdsplusgval.f	. 2 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
21		prdsplusgval.g	. 2 ⊢ (𝜑 → 𝐺 ∈ 𝐵)
22		xrltso 12875	. . . 4 ⊢ < Or ℝ^*
23	22	supex 9222	. . 3 ⊢ sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ) ∈ V
24	23	a1i 11	. 2 ⊢ (𝜑 → sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ) ∈ V)
25	11, 19, 20, 21, 24	ovmpod 7425	1 ⊢ (𝜑 → (𝐹𝐷𝐺) = sup((ran (𝑥 ∈ 𝐼 ↦ ((𝐹‘𝑥)(dist‘(𝑅‘𝑥))(𝐺‘𝑥))) ∪ {0}), ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ∪ cun 3885 {csn 4561 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 supcsup 9199 0cc0 10871 ℝ^*cxr 11008 < clt 11009 Basecbs 16912 distcds 16971 X_scprds 17156

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-fz 13240 df-struct 16848 df-slot 16883 df-ndx 16895 df-base 16913 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-hom 16986 df-cco 16987 df-prds 17158

This theorem is referenced by: prdsdsval2 17195 xpsdsval 23534

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