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Theorem dvivthlem2 26183

Description: Lemma for dvivth 26184. (Contributed by Mario Carneiro, 20-Feb-2015.)

**Hypotheses**
Ref	Expression
dvivth.1	⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵))
dvivth.2	⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵))
dvivth.3	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
dvivth.4	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
dvivth.5	⊢ (𝜑 → 𝑀 < 𝑁)
dvivth.6	⊢ (𝜑 → 𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)))
dvivth.7	⊢ 𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦)))

**Assertion**
Ref	Expression
dvivthlem2	⊢ (𝜑 → 𝐶 ∈ ran (ℝ D 𝐹))

Distinct variable groups: 𝑦,𝐴 𝑦,𝐵 𝑦,𝐹 𝑦,𝑀 𝑦,𝐶 𝑦,𝑁 𝜑,𝑦 Allowed substitution hint: 𝐺(𝑦)

Proof of Theorem dvivthlem2 Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvivth.1	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵))
2		dvivth.2	. . 3 ⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵))
3		dvivth.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
4		dvivth.4	. . 3 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
5		dvivth.5	. . 3 ⊢ (𝜑 → 𝑀 < 𝑁)
6		dvivth.6	. . 3 ⊢ (𝜑 → 𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)))
7		dvivth.7	. . 3 ⊢ 𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦)))
8	1, 2, 3, 4, 5, 6, 7	dvivthlem1 26182	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶)
9		dvf 26081	. . . . . . 7 ⊢ (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ
10	4	feq2d 6693	. . . . . . 7 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℂ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ))
11	9, 10	mpbii 236	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℂ)
12	11	ffnd 6710	. . . . 5 ⊢ (𝜑 → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
13		iccssioo2 13456	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐴(,)𝐵) ∧ 𝑁 ∈ (𝐴(,)𝐵)) → (𝑀[,]𝑁) ⊆ (𝐴(,)𝐵))
14	1, 2, 13	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ (𝐴(,)𝐵))
15	14	sselda 3940	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝑥 ∈ (𝐴(,)𝐵))
16		fnfvelrn 7079	. . . . 5 ⊢ (((ℝ D 𝐹) Fn (𝐴(,)𝐵) ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ran (ℝ D 𝐹))
17	12, 15, 16	syl2an2r 698	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐹)‘𝑥) ∈ ran (ℝ D 𝐹))
18		eleq1 2854	. . . 4 ⊢ (((ℝ D 𝐹)‘𝑥) = 𝐶 → (((ℝ D 𝐹)‘𝑥) ∈ ran (ℝ D 𝐹) ↔ 𝐶 ∈ ran (ℝ D 𝐹)))
19	17, 18	syl5ibcom 248	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (((ℝ D 𝐹)‘𝑥) = 𝐶 → 𝐶 ∈ ran (ℝ D 𝐹)))
20	19	rexlimdva 3169	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶 → 𝐶 ∈ ran (ℝ D 𝐹)))
21	8, 20	mpd 16	1 ⊢ (𝜑 → 𝐶 ∈ ran (ℝ D 𝐹))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ∃wrex 3092 ⊆ wss 3908 class class class wbr 5112 ↦ cmpt 5195 dom cdm 5664 ran crn 5665 Fn wfn 6535 ⟶wf 6536 ‘cfv 6540 (class class class)co 7416 ℂcc 11108 ℝcr 11109 · cmul 11115 < clt 11253 − cmin 11451 (,)cioo 13382 [,]cicc 13385 –cn→ccncf 25050 D cdv 26037

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5241 ax-sep 5260 ax-nul 5272 ax-pow 5339 ax-pr 5407 ax-un 7738 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4491 df-pw 4567 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4876 df-int 4916 df-iun 4961 df-iin 4962 df-br 5113 df-opab 5177 df-mpt 5196 df-tr 5222 df-id 5559 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-se 5618 df-we 5619 df-xp 5670 df-rel 5671 df-cnv 5672 df-co 5673 df-dm 5674 df-rn 5675 df-res 5676 df-ima 5677 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7680 df-om 7865 df-1st 7988 df-2nd 7989 df-supp 8159 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-2o 8456 df-er 8696 df-map 8828 df-pm 8829 df-ixp 8898 df-en 8946 df-dom 8947 df-sdom 8948 df-fin 8949 df-fsupp 9324 df-fi 9373 df-sup 9404 df-inf 9405 df-oi 9474 df-card 9936 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11453 df-neg 11454 df-div 11882 df-nn 12244 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12515 df-z 12602 df-dec 12722 df-uz 12873 df-q 12983 df-rp 13027 df-xneg 13147 df-xadd 13148 df-xmul 13149 df-ioo 13386 df-ico 13388 df-icc 13389 df-fz 13546 df-fzo 13694 df-seq 14049 df-exp 14109 df-hash 14378 df-cj 15161 df-re 15162 df-im 15163 df-sqrt 15297 df-abs 15298 df-struct 17217 df-sets 17234 df-slot 17252 df-ndx 17264 df-base 17280 df-ress 17301 df-plusg 17333 df-mulr 17334 df-starv 17335 df-sca 17336 df-vsca 17337 df-ip 17338 df-tset 17339 df-ple 17340 df-ds 17342 df-unif 17343 df-hom 17344 df-cco 17345 df-rest 17485 df-topn 17486 df-0g 17504 df-gsum 17505 df-topgen 17506 df-pt 17507 df-prds 17510 df-xrs 17566 df-qtop 17571 df-imas 17572 df-xps 17574 df-mre 17648 df-mrc 17649 df-acs 17651 df-mgm 18708 df-sgrp 18787 df-mnd 18803 df-submnd 18852 df-mulg 19144 df-cntz 19397 df-cmn 19862 df-psmet 21529 df-xmet 21530 df-met 21531 df-bl 21532 df-mopn 21533 df-fbas 21534 df-fg 21535 df-cnfld 21538 df-top 23066 df-topon 23083 df-topsp 23105 df-bases 23118 df-cld 23191 df-ntr 23192 df-cls 23193 df-nei 23270 df-lp 23308 df-perf 23309 df-cn 23399 df-cnp 23400 df-haus 23487 df-cmp 23559 df-tx 23734 df-hmeo 23927 df-fil 24018 df-fm 24110 df-flim 24111 df-flf 24112 df-xms 24492 df-ms 24493 df-tms 24494 df-cncf 25052 df-limc 26040 df-dv 26041

This theorem is used by: dvivth 26184

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